<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">1. Motivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Trig substitution</h3>
<div
id="video_technique2-tab1-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:h49BDpcooZA", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab1-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab1-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab1-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab1-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">2. Trig substitution</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab2-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab2-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Use <span style="color:#27408C"><b class="bf">trigonometric substitution</b></span> to evaluate integrands that contain [mathjaxinline]\displaystyle \sqrt {a^2-x^2}\,[/mathjaxinline], [mathjaxinline]\displaystyle \sqrt {a^2+x^2}\,[/mathjaxinline], and [mathjaxinline]\displaystyle \sqrt {x^2-a^2}\, .[/mathjaxinline] </p></li><li><p>
Use the method of <span style="color:#27408C"><b class="bf">completing the square</b></span> to manipulate integrals into one of the above forms, and perform a trig substitution. </p></li></ul><p><b class="bfseries">Contents: 10 pages</b></p><p>
7 videos (57 minutes 1x speed) 13 questions </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab3" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">3. Motivating example</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab3-problem1" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab3-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="2"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab3-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1-problem-progress" tabindex="-1">
Set up the integral
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab3-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
In this section, we will learn the technique of trigonometric substitution to evaluate integrals such as the one below.<br/></p>
<center>
<img alt="A circle with radius a centered at the origin is plotted in the x y plane. The value b is indicated on the positive x axis. The shaded region capital A is the region bounded by the y axis, the vertical line x equals b, and the top of the circle." src="/assets/courseware/v1/526f103fa0900d680fd278b51a58dd28/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_areaincircletab.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
<br/>
</center>
<p>
Set up the integral for the area of the shaded region using vertical slices.<br/></p>
<span>
<style>
.xmodule_display.xmodule_CapaModule .problem .capa_inputtype.textline input {
min-width: 0 !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
display: inline-block !important;
}
.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax_Preview {
display: inline-block !important;
}
</style>
<table>
<tbody>
<tr>
<td/>
<td colspan="4">
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_technique2-tab3-problem1_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_technique2-tab3-problem1_2_1" id="input_technique2-tab3-problem1_2_1" aria-describedby="status_technique2-tab3-problem1_2_1" value="" size="5"/>
<span class="trailing_text" id="trailing_text_technique2-tab3-problem1_2_1"/>
<span class="status unanswered" id="status_technique2-tab3-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab3-problem1_2_1" class="answer"/>
</div>
</div></div>
</td>
</tr>
<tr>
<td>
<p style="text-align:left"> \(\displaystyle \Large{A \,=\,} \)</p>
</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique2-tab3-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab3-problem1_3_1" id="input_technique2-tab3-problem1_3_1" data-input-id="technique2-tab3-problem1_3_1" value="" aria-describedby="trailing_text_technique2-tab3-problem1_3_1 status_technique2-tab3-problem1_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab3-problem1_3_1"> [mathjaxinline] \Large{dx} [/mathjaxinline]</span>
<span class="status unanswered" id="status_technique2-tab3-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab3-problem1_3_1" class="answer"/>
<div id="input_technique2-tab3-problem1_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</td>
</tr>
<tr>
<td/>
<td colspan="4">
<p style="display:inline; text-align:right"> \( \displaystyle \large{ 0}\)</p>
</td>
</tr>
</tbody>
</table>
</span>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab3-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Set up the integral" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab3-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab3-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab3-problem1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab3-problem1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab3-problem1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab4" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">4. Trig substitution example</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Worked example</h3>
<div
id="video_technique2-tab4-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:cO3aANUG-gk", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab4-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab4-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab4-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab4-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab4-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab4-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Worked example with trig substitution</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Step</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Consider the integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int \frac{dx}{x^2\sqrt {1+x^2}}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Identify ugliest part of integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {1+x^2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) <table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none; border:none">
Make a substitution to simplify ugly expression:</td></tr><tr><td style="text-align:left; border:none; border:none">
Based on [mathjaxinline]\displaystyle \sec ^2\theta = 1+\tan ^2\theta[/mathjaxinline]. </td></tr></table></td><td style="text-align:left; border:none">
[mathjaxinline]\begin{cases} x= \tan \theta \\ \sqrt {1+x^2} = \sqrt {\sec ^2\theta } = \sec \theta \\ dx = \sec ^2\theta \, d\theta \end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(Note that generally, [mathjaxinline]\sqrt {\sec ^2\theta } = |\sec \theta |[/mathjaxinline]. However, when we make the substituion [mathjaxinline]x = \tan \theta[/mathjaxinline], we are implicitly restricting to the range [mathjaxinline]-\pi /2 < \theta < \pi /2[/mathjaxinline]. On this domain, [mathjaxinline]x[/mathjaxinline] achieves all real values, [mathjaxinline]\tan (\theta )[/mathjaxinline] is continuous and invertible, and [mathjaxinline]\sec (\theta )>0[/mathjaxinline].) (4) Plug into integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int \frac{dx}{x^2\sqrt {1+x^2}} = \int \frac{\sec \theta \, d\theta }{\tan ^2 \theta }[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(5) Rewrite in terms of [mathjaxinline]\sin[/mathjaxinline] and [mathjaxinline]\cos[/mathjaxinline]: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \int \frac{\cos \theta \, d\theta }{\sin ^2 \theta }[/mathjaxinline] </td></tr><tr><td style="text-align:left; border:none">
(6) Still not done. Need to evaluate this integral. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
How about trying a direct substitution? </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \begin{cases} u = \sin \theta \\ u^2 = \sin ^2 \theta \\ du = \cos \theta \, d\theta \\ \end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \int \frac{du}{u^2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(7) Evaluate the integral. </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{-1}{u}+C[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
This is the end of the calculus. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
(8) The rest is back substitution. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
First replace [mathjaxinline]u[/mathjaxinline] by [mathjaxinline]\sin \theta[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]= \displaystyle \frac{-1}{\sin \theta } +C[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(9) Draw a diagram of [mathjaxinline]x = \tan \theta[/mathjaxinline]: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/507e9132ced4fad75caeed675fc20201/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri1.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled x and the side adjacent to theta is labeled 1." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(10) Use diagram to determine [mathjaxinline]\displaystyle \frac{1}{\sin \theta }[/mathjaxinline] in terms of [mathjaxinline]x[/mathjaxinline]: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/26895bb377983c388dc060d1d25a3ae5/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri2.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled x and the side adjacent to theta is labeled 1. The hypotenuse is labeled square root of 1 plus x squared." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{\sin \theta } = \frac{\sqrt {1+x^2}}{x}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(11) Rewrite in terms of [mathjaxinline]x[/mathjaxinline]: </td><td style="text-align:left; border:none">
[mathjaxinline]= \displaystyle -\frac{\sqrt {1+x^2}}{x} + C[/mathjaxinline] </td></tr></table><p>
The non-calculus part of this technique, the part of undoing the trig substitution, requires some practice. Note that in Steps (8), (9), and (10) we are actually computing the trig function of an inverse trig function. In our example, we had </p><table id="a0000002124" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002125"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \theta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \arctan x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.286)</td></tr><tr id="a0000002126"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \csc \theta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \csc \left(\arctan x \right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.287)</td></tr></table><p>
Let's get some practice with this non-calculus component of the method now. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab5" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">5. Undoing trig substitutions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab5-problem1" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab5-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab5-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1-problem-progress" tabindex="-1">
Review inverse trig functions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
What is the customary range of [mathjaxinline]\theta = \arctan x[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_technique2-tab5-problem1_2_1">
<fieldset aria-describedby="status_technique2-tab5-problem1_2_1">
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_2_1" id="input_technique2-tab5-problem1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="technique2-tab5-problem1_2_1-choice_1-label" for="input_technique2-tab5-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_2_1">
<text> [mathjaxinline]0 &lt; \theta &lt; \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_2_1" id="input_technique2-tab5-problem1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="technique2-tab5-problem1_2_1-choice_2-label" for="input_technique2-tab5-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_2_1">
<text> [mathjaxinline]\displaystyle \frac{-\pi }{2} &lt; \theta &lt; \frac{\pi }{2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_2_1" id="input_technique2-tab5-problem1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="technique2-tab5-problem1_2_1-choice_3-label" for="input_technique2-tab5-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_2_1">
<text> [mathjaxinline]-\pi &lt; \theta &lt; \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_2_1" id="input_technique2-tab5-problem1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="technique2-tab5-problem1_2_1-choice_4-label" for="input_technique2-tab5-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_2_1">
<text> [mathjaxinline]0 &lt; \theta &lt; 2\pi[/mathjaxinline]</text>
</label>
</div>
<span id="answer_technique2-tab5-problem1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_technique2-tab5-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
What is the customary range of [mathjaxinline]\theta = \arcsin x[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_technique2-tab5-problem1_3_1">
<fieldset aria-describedby="status_technique2-tab5-problem1_3_1">
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_3_1" id="input_technique2-tab5-problem1_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="technique2-tab5-problem1_3_1-choice_1-label" for="input_technique2-tab5-problem1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_3_1">
<text> [mathjaxinline]0 \leq \theta \leq \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_3_1" id="input_technique2-tab5-problem1_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="technique2-tab5-problem1_3_1-choice_2-label" for="input_technique2-tab5-problem1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_3_1">
<text> [mathjaxinline]\displaystyle \frac{-\pi }{2} \leq \theta \leq \frac{\pi }{2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_3_1" id="input_technique2-tab5-problem1_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="technique2-tab5-problem1_3_1-choice_3-label" for="input_technique2-tab5-problem1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_3_1">
<text> [mathjaxinline]-\pi \leq \theta \leq \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_3_1" id="input_technique2-tab5-problem1_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="technique2-tab5-problem1_3_1-choice_4-label" for="input_technique2-tab5-problem1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_3_1">
<text> [mathjaxinline]0 \leq \theta \leq 2\pi[/mathjaxinline]</text>
</label>
</div>
<span id="answer_technique2-tab5-problem1_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_technique2-tab5-problem1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
What is the customary range of [mathjaxinline]\theta = \arccos x[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_technique2-tab5-problem1_4_1">
<fieldset aria-describedby="status_technique2-tab5-problem1_4_1">
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_4_1" id="input_technique2-tab5-problem1_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="technique2-tab5-problem1_4_1-choice_1-label" for="input_technique2-tab5-problem1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_4_1">
<text> [mathjaxinline]0 \leq \theta \leq \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_4_1" id="input_technique2-tab5-problem1_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="technique2-tab5-problem1_4_1-choice_2-label" for="input_technique2-tab5-problem1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_4_1">
<text> [mathjaxinline]\displaystyle \frac{-\pi }{2} \leq \theta \leq \frac{\pi }{2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_4_1" id="input_technique2-tab5-problem1_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="technique2-tab5-problem1_4_1-choice_3-label" for="input_technique2-tab5-problem1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_4_1">
<text> [mathjaxinline]-\pi \leq \theta \leq \pi[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_technique2-tab5-problem1_4_1" id="input_technique2-tab5-problem1_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="technique2-tab5-problem1_4_1-choice_4-label" for="input_technique2-tab5-problem1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_technique2-tab5-problem1_4_1">
<text> [mathjaxinline]0 \leq \theta \leq 2\pi[/mathjaxinline]</text>
</label>
</div>
<span id="answer_technique2-tab5-problem1_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_technique2-tab5-problem1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_technique2-tab5-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Review inverse trig functions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab5-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab5-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Undoing trig substitutions</h3>
<div
id="video_technique2-tab5-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:lqyYGoN_iLc", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab5-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab5-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab5-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab5-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab5-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab5-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
To undo a trig substitution, you must often evaluate a function of the form a trig function of an inverse trig function. For example: </p><ul class="itemize"><li><p>
[mathjaxinline]\csc \left(\arctan x \right)[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\tan \left(\mathrm{arccsc} x\right)[/mathjaxinline] </p></li></ul><p>
To do this the steps are as follows: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Undoing a trig substitution</b></td><td style="text-align:left; border:none"><b class="bf">Worked example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Compute the composite: </td><td style="text-align:left; border:none">
[mathjaxinline]\tan \underbrace{\left( \mathrm{arccsc}(x)\right)}_{\theta }[/mathjaxinline].</td></tr><tr><td style="text-align:left; border:none">
(2) Draw a picture of </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\begin{array}{ll}& \theta = \mathrm{arccsc}(x) \\ \Longleftrightarrow & \csc \theta = x\\ \Longleftrightarrow & \displaystyle \sin \theta = \frac{1}{x} \end{array}[/mathjaxinline] </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/89719413b0f8ed23b1033743c9c3c031/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri0.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled 1 and the hypotenuse is labeled x." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(3) Find length of remaining side: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/8088615ba41729fdb262399606c86234/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri0b.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled 1 and the hypotenuse is labeled x. The side adjacent to theta is labeled square root of x squared minus 1." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(4) Compute trig function: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \tan \theta = \frac{1}{\sqrt {x^2-1}}[/mathjaxinline] </td></tr></table><p>
From the pictures and formulas, we can see that we used [mathjaxinline]1 < x < \infty[/mathjaxinline] and [mathjaxinline]0 < \theta < \pi /2[/mathjaxinline]. </p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab5-problem2" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab5-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="4"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab5-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2-problem-progress" tabindex="-1">
Practice with trig functions of inverse trig functions
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab5-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \sin \left(\arccos (x)\right)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab5-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab5-problem2_2_1" id="input_technique2-tab5-problem2_2_1" data-input-id="technique2-tab5-problem2_2_1" value="" aria-describedby="status_technique2-tab5-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab5-problem2_2_1"/>
<span class="status unanswered" id="status_technique2-tab5-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab5-problem2_2_1" class="answer"/>
<div id="input_technique2-tab5-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \tan \left(\mathrm{arcsec}(x)\right)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique2-tab5-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab5-problem2_3_1" id="input_technique2-tab5-problem2_3_1" data-input-id="technique2-tab5-problem2_3_1" value="" aria-describedby="status_technique2-tab5-problem2_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab5-problem2_3_1"/>
<span class="status unanswered" id="status_technique2-tab5-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab5-problem2_3_1" class="answer"/>
<div id="input_technique2-tab5-problem2_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \sec \left(\arcsin (x/a)\right)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_technique2-tab5-problem2_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab5-problem2_4_1" id="input_technique2-tab5-problem2_4_1" data-input-id="technique2-tab5-problem2_4_1" value="" aria-describedby="status_technique2-tab5-problem2_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab5-problem2_4_1"/>
<span class="status unanswered" id="status_technique2-tab5-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab5-problem2_4_1" class="answer"/>
<div id="input_technique2-tab5-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \sin \left(\arctan (x/a)\right)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_technique2-tab5-problem2_5_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab5-problem2_5_1" id="input_technique2-tab5-problem2_5_1" data-input-id="technique2-tab5-problem2_5_1" value="" aria-describedby="status_technique2-tab5-problem2_5_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab5-problem2_5_1"/>
<span class="status unanswered" id="status_technique2-tab5-problem2_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab5-problem2_5_1" class="answer"/>
<div id="input_technique2-tab5-problem2_5_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab5-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice with trig functions of inverse trig functions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab5-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab5-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab5-problem2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab6" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">6. Preparation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab6-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab6-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
When you use a trig substitution to solve an integral, the reverse process involves taking an inverse trig function. To help determine when you should use a trig substitution, we want you to recall the derivatives of these inverse trig functions below. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab6-problem1" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab6-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab6-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1-problem-progress" tabindex="-1">
Derivatives of inverse trig functions 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate the derivative (assume [mathjaxinline]a&gt;0[/mathjaxinline]). </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dx}\arcsin (x/a) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab6-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab6-problem1_2_1" id="input_technique2-tab6-problem1_2_1" data-input-id="technique2-tab6-problem1_2_1" value="" aria-describedby="status_technique2-tab6-problem1_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_technique2-tab6-problem1_2_1"/>
<span class="status unanswered" id="status_technique2-tab6-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab6-problem1_2_1" class="answer"/>
<div id="input_technique2-tab6-problem1_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab6-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Derivatives of inverse trig functions 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab6-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab6-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab6-problem2" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab6-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab6-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2-problem-progress" tabindex="-1">
Derivatives of inverse trig functions 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate the derivative (assume [mathjaxinline]a&gt;0[/mathjaxinline]). </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dx}\arctan \left(x/a\right) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab6-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab6-problem2_2_1" id="input_technique2-tab6-problem2_2_1" data-input-id="technique2-tab6-problem2_2_1" value="" aria-describedby="status_technique2-tab6-problem2_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_technique2-tab6-problem2_2_1"/>
<span class="status unanswered" id="status_technique2-tab6-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab6-problem2_2_1" class="answer"/>
<div id="input_technique2-tab6-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab6-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Derivatives of inverse trig functions 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab6-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab6-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab6-problem3" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab6-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab6-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3-problem-progress" tabindex="-1">
Derivatives of inverse trig functions 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab6-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate the derivative (assume [mathjaxinline]a&gt;0[/mathjaxinline]). </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dx}\mathrm{arcsec}(x/a) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab6-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab6-problem3_2_1" id="input_technique2-tab6-problem3_2_1" data-input-id="technique2-tab6-problem3_2_1" value="" aria-describedby="status_technique2-tab6-problem3_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_technique2-tab6-problem3_2_1"/>
<span class="status unanswered" id="status_technique2-tab6-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab6-problem3_2_1" class="answer"/>
<div id="input_technique2-tab6-problem3_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab6-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Derivatives of inverse trig functions 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab6-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab6-problem3">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem3-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem3-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab6-problem3-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab7" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">7. Method of trig substitution</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Summary of trig substitution</h3>
<div
id="video_technique2-tab7-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:HxNm2A4Rveo", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab7-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab7-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab7-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab7-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab7-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab7-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Summary of trig substitutions</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">If integrand contains: </b></td><td style="text-align:left; border:none"><b class="bf">Make substitution:</b></td><td style="text-align:left; border:none"><b class="bf">To get:</b></td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2-x^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = \begin{cases} a\cos (\theta )\\ a\sin (\theta )\end{cases}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2-x^2}= \begin{cases} a\sin (\theta ) \\ \displaystyle a\cos \theta )\end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2+x^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = a\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2+x^2}= a\sec (\theta )[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {x^2-a^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = a\sec (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {x^2-a^2}=a\tan (\theta )[/mathjaxinline] </td></tr></table>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab7-problem1" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab7-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab7-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1-problem-progress" tabindex="-1">
Practice trig subs 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem1-problem-progress"></div>
<div class="problem">
<div>
<center>
<img alt="A circle with radius a centered at the origin is plotted in the x y plane. The value b is indicated on the positive x axis. The shaded region capital A is the region bounded by the y axis, the vertical line x equals b, and the top of the circle." src="/assets/courseware/v1/526f103fa0900d680fd278b51a58dd28/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_areaincircletab.svg" style="margin: 10px 25px 25px 25px" width="300px"/>
<br/>
</center>
<p>
Recall the area [mathjaxinline]\, A\,[/mathjaxinline] of the &#8220;tab" in a circle shaded in the figure above is given by [mathjaxinline]\, \displaystyle \int _0^ b \, \sqrt {a^2-x^2}\, dx[/mathjaxinline].<br/></p>
<p>
Evaluate the integral.<br/>(Enter your answer in terms of [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline].)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\, \, \displaystyle \int _0^ b \, \sqrt {a^2-x^2}\, dx \, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab7-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab7-problem1_2_1" id="input_technique2-tab7-problem1_2_1" data-input-id="technique2-tab7-problem1_2_1" value="" aria-describedby="status_technique2-tab7-problem1_2_1" size="70"/>
<span class="trailing_text" id="trailing_text_technique2-tab7-problem1_2_1"/>
<span class="status unanswered" id="status_technique2-tab7-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab7-problem1_2_1" class="answer"/>
<div id="input_technique2-tab7-problem1_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab7-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice trig subs 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab7-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab7-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab7-problem2" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab7-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab7-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2-problem-progress" tabindex="-1">
Practice trig subs 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Evaluate [mathjaxinline]\, \displaystyle \int \frac{dx}{\sqrt {\left(2x^2+18\right)^3}}[/mathjaxinline].<br/>(Use [mathjaxinline]\, C\,[/mathjaxinline] as the constant of integration.)<br/>(Do not include inverse trigonometric functions in your answer.)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\, \, \displaystyle \int \frac{dx}{\sqrt {\left(2x^2+18\right)^3}}\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab7-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab7-problem2_2_1" id="input_technique2-tab7-problem2_2_1" data-input-id="technique2-tab7-problem2_2_1" value="" aria-describedby="status_technique2-tab7-problem2_2_1" size="70"/>
<span class="trailing_text" id="trailing_text_technique2-tab7-problem2_2_1"/>
<span class="status unanswered" id="status_technique2-tab7-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab7-problem2_2_1" class="answer"/>
<div id="input_technique2-tab7-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab7-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice trig subs 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab7-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab7-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab7-problem3" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab7-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab7-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3-problem-progress" tabindex="-1">
Practice trig subs 3 (*)
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab7-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Evaluate [mathjaxinline]\, \displaystyle \int _{-1}^{-1/2} \frac{dx}{\sqrt {4x^2-1}}[/mathjaxinline].<br/>(You can enter the functional form of your numerical answer, e.g. you can type &#8220;tan(pi/3)" for [mathjaxinline]\tan (\pi /3)[/mathjaxinline].)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\, \, \displaystyle \int _{-1}^{-1/2} \frac{dx}{\sqrt {4x^2-1}}\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab7-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab7-problem3_2_1" id="input_technique2-tab7-problem3_2_1" data-input-id="technique2-tab7-problem3_2_1" value="" aria-describedby="status_technique2-tab7-problem3_2_1" size="70"/>
<span class="trailing_text" id="trailing_text_technique2-tab7-problem3_2_1"/>
<span class="status unanswered" id="status_technique2-tab7-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab7-problem3_2_1" class="answer"/>
<div id="input_technique2-tab7-problem3_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab7-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice trig subs 3 (*)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab7-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab7-problem3">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem3-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem3-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab7-problem3-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab8" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">8. Completing the square</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab8-problem1" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab8-problem1-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab8-problem1-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1-problem-progress" tabindex="-1">
Derivatives whose antiderivatives require this technique
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem1-problem-progress"></div>
<div class="problem">
<div>
<p>
Calculate the derivative </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dx}\arcsin (\sqrt {1-x}) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab8-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem1_2_1" id="input_technique2-tab8-problem1_2_1" data-input-id="technique2-tab8-problem1_2_1" value="" aria-describedby="status_technique2-tab8-problem1_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem1_2_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem1_2_1" class="answer"/>
<div id="input_technique2-tab8-problem1_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div class="mymodal-wrap" style="width: 185px; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">
<a class="mymodal-btn mymodal-btn-big" href="#mymodal-one">FORMULA INPUT HELP
</a>
</div>
<div class="mymodal-positioner">
<a aria-hidden="true" class="mymodal" href="#" id="mymodal-one"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Formula Input Guide</h4>
<a class="mymodal-btn-close" href="#">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a class="mymodal-btn" href="#">Close Guide</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_technique2-tab8-problem1_solution_1"/>
</div></p>
<script src="/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/st.js" type="text/javascript"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Derivatives whose antiderivatives require this technique" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab8-problem1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab8-problem1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Completing the square example</h3>
<div
id="video_technique2-tab8-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:CoF95PY7Jq0", "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/e8f5f9ed3a07496f95f92bf0ce1d2b48.jpeg?AWSAccessKeyId=ASIA4QDFFSUR6KTXH6OY&Signature=%2BWf%2F7BcKgShpsqx7gSPLinfoVtI%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEMX%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIQCyIjBegnIyDooI2xA%2FVlp2B2795uQKI7QZamVBVjMVYAIgcVmXTpY1aKUJwmYQM1yIBH4vc5OuRg6d84sA4l4KZw0qwgUI%2Fv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARADGgw4NTkyMDU2Mzc0MTEiDGB3S5D16ogPhv7IISqWBZxHk4ZGSW2BPPO%2FIQxH5HGcPlikyRNSC%2Febrkle1W1GYcWlXpZJUocutjl5SQImuwR98tRiBexotWeHu6LirvN9zFvMIJg7lovfW%2BxOddtvxvM10%2Br%2B6j8q%2Bt8tR7yddFoNLKHVL1Qx%2BUBTO%2F2X7iHJVgLOi6UGjuO701tDWj3jDCrRK4%2BP2LeVu0M9OseCnAJ0C9a4yKlJoQpPB38Nz61m8zHnG0jCfDj%2FPejjXmJy7pslvq9COYUb99v8Z0TfT%2B5VFORFCDETOPh%2FBocxybY640yinH%2BweFP8OzZaLI%2B658Du7cxhNBU%2B6JfmQPRJL78GsQlJHsFlUc4jlMjUmtSloBE5PFE1reIM0awmgGyh%2BsRkww6c6btNe8XT5MsUjfVcUNuLFxX5vCCwsQTLhR2GSi9GyiZZxe%2BEGyR%2BV0qghHmC%2BItHfl7HvYKR6hqLTOq9Fy%2Bd%2BiCByjX3ZR8kxfkthegwTOcemk0VW5PXh7GxvPtVrbCYILrPnJvWFsdiIgW%2BUDA9fCDj9zAemR1S6gv%2BF%2BAzP8Y01b%2FWLtG6Jhkd%2Bhk6m21%2F6w7aib02E%2BeZpYW%2F%2B4p7jPbn4KJuSDfz2DZadykUIGe3oaVupl%2BSTTqgXlzZzl2Wz6WnH31ZX5MWZbDFQFTXi6JE2dOTfI2IHGkG9HC7EIEJE%2FmoJeLuuTYGaHTeyHjHM51%2Brdsb6yWcGvUsX99GEj2vwa1WUt%2BrAPTsIsjrFY%2BfzebDy652WA486ZgEm4wMQ72tWDMwXu3QzKmOvNJv9Ur2sF%2BOexG5XHXzBUKaxPSJMaq7yDDIrWex4IX4578uXZHZvwZ0HS35QZaMq5t3Ii60O4sUutkInVLc6lYys%2BByiaMWkLSabzzw8wZCWDKUMMnlh7EGOrEBeLZsCmXc%2FbrSAlfyTZE105%2B7UfIxvtkmheZIQyrZby9VOvWh2TOw9nDxWi7jpceuWmKGGgyVambXbx0YKWNs%2BRvkB2JmDnbcMp64hauDL5d9F%2FHZSrBkYwUrkxoJEg28Z64x4h%2Ff%2B62p4Tbp2ZRQx5ZZSuopnkyR5izMBJiFGQzOT1VVfK%2FL07Ui8g8tl6%2F0qUqUJ7eaVRw8wRZ%2Bf28WbNqbNRFNehGud1PZDJXgewPo&Expires=1713503756", "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT1801X/MIT1801XT314-V033300_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT1801XT314-V033300/MIT1801XT314-V033300.m3u8"], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 639.01}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab8-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab8-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab8-video1">
<div class="wrapper-download-video">
<h4 class="hd hd-5">Video</h4>
<a class="btn-link video-sources video-download-button" href="https://d2f1egay8yehza.cloudfront.net/MIT1801X/MIT1801XT314-V033300_DTH.mp4">
Download video file
</a>
</div>
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab8-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab8-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bf">Video note:</b> There is an absolute value sign missing inside the logarithm at 8:38 onwards.<br/></p><p><b class="bfseries">Completing the square</b></p><p>
We use completing the square to rewrite any quadratic function into a form which is more conducive to using trigonometric substitution.<br/></p><p>
To complete the square means to rewrite any quadratic function [mathjaxinline]\, Ax^2+Bx+C\,[/mathjaxinline] into the form [mathjaxinline]\, A(x+b)^2+c[/mathjaxinline].<br/></p><p>
Here is the algebra:<br/></p><table id="a0000002215" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002216"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle Ax^2+Bx+C[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x^2+\frac{B}{A}x \right)+C \qquad \text {Factor out }\, A[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.346)</td></tr><tr id="a0000002217"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Add and subtract the square term}\qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x^2+\frac{B}{A}x+ \left(\frac{B}{2A} \right)^2 \right) - A\left(\frac{B}{2A} \right)^2+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.347)</td></tr><tr id="a0000002218"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x+\frac{B}{2A} \right)^2 +\left(-\frac{B^2}{4A}+C\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.348)</td></tr></table>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab8-problem2" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab8-problem2-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab8-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2-problem-progress" tabindex="-1">
Practice completing the square 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem2-problem-progress"></div>
<div class="problem">
<div>
<p>
Find [mathjaxinline]\, a,\,[/mathjaxinline] [mathjaxinline]\, b,\,[/mathjaxinline] [mathjaxinline]\, c\,[/mathjaxinline] such that [mathjaxinline]\, \displaystyle \frac{x^2}{9}-\frac{4}{3}x-5\, =\, a\, (x+b)^2+c[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\, \, a\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab8-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem2_2_1" id="input_technique2-tab8-problem2_2_1" data-input-id="technique2-tab8-problem2_2_1" value="" aria-describedby="status_technique2-tab8-problem2_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem2_2_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem2_2_1" class="answer"/>
<div id="input_technique2-tab8-problem2_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\, \, b\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique2-tab8-problem2_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem2_3_1" id="input_technique2-tab8-problem2_3_1" data-input-id="technique2-tab8-problem2_3_1" value="" aria-describedby="status_technique2-tab8-problem2_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem2_3_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem2_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem2_3_1" class="answer"/>
<div id="input_technique2-tab8-problem2_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\, \, c\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_technique2-tab8-problem2_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem2_4_1" id="input_technique2-tab8-problem2_4_1" data-input-id="technique2-tab8-problem2_4_1" value="" aria-describedby="status_technique2-tab8-problem2_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem2_4_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem2_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem2_4_1" class="answer"/>
<div id="input_technique2-tab8-problem2_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<div class="solution-span">
<span id="solution_technique2-tab8-problem2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Practice completing the square 1" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab8-problem2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab8-problem2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem2-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem2-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem2-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab8-problem3" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab8-problem3-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="3"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab8-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3-problem-progress" tabindex="-1">
Complete the square 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem3-problem-progress"></div>
<div class="problem">
<div>
<p>
Find [mathjaxinline]\, a,\,[/mathjaxinline] [mathjaxinline]\, b,\,[/mathjaxinline] [mathjaxinline]\, c\,[/mathjaxinline] such that [mathjaxinline]\, x(x+8)= a\, (x+b)^2+c[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\, \, a\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab8-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem3_2_1" id="input_technique2-tab8-problem3_2_1" data-input-id="technique2-tab8-problem3_2_1" value="" aria-describedby="status_technique2-tab8-problem3_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem3_2_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem3_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem3_2_1" class="answer"/>
<div id="input_technique2-tab8-problem3_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\, \, b\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique2-tab8-problem3_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem3_3_1" id="input_technique2-tab8-problem3_3_1" data-input-id="technique2-tab8-problem3_3_1" value="" aria-describedby="status_technique2-tab8-problem3_3_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem3_3_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem3_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem3_3_1" class="answer"/>
<div id="input_technique2-tab8-problem3_3_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<p style="display:inline">[mathjaxinline]\, \, c\, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_technique2-tab8-problem3_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem3_4_1" id="input_technique2-tab8-problem3_4_1" data-input-id="technique2-tab8-problem3_4_1" value="" aria-describedby="status_technique2-tab8-problem3_4_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem3_4_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem3_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem3_4_1" class="answer"/>
<div id="input_technique2-tab8-problem3_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<div class="solution-span">
<span id="solution_technique2-tab8-problem3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Complete the square 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab8-problem3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab8-problem3">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem3-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem3-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem3-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Worked example: trig substitution using completing the square</h3>
<div
id="video_technique2-tab8-video2"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:9Ou0S_MvQA4", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab8-video2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab8-video2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab8-video2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab8-video2/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="problem" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="True" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_technique2-tab8-problem4" class="problems-wrapper" role="group"
aria-labelledby="technique2-tab8-problem4-problem-title"
data-problem-id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4" data-url="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="1"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="technique2-tab8-problem4-problem-title" aria-describedby="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4-problem-progress" tabindex="-1">
Integrate
</h3>
<div class="problem-progress" id="block-v1:MITx+18.01.2x+3T2019+type@problem+block@technique2-tab8-problem4-problem-progress"></div>
<div class="problem">
<div>
<p>
Use your answer from above to evaluate the integral [mathjaxinline]\, \displaystyle \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt {x(x+8)}}\, dx[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\, \, \displaystyle \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt {x(x+8)}}\, dx \, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique2-tab8-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique2-tab8-problem4_2_1" id="input_technique2-tab8-problem4_2_1" data-input-id="technique2-tab8-problem4_2_1" value="" aria-describedby="status_technique2-tab8-problem4_2_1" size="20"/>
<span class="trailing_text" id="trailing_text_technique2-tab8-problem4_2_1"/>
<span class="status unanswered" id="status_technique2-tab8-problem4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_technique2-tab8-problem4_2_1" class="answer"/>
<div id="input_technique2-tab8-problem4_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
<div class="solution-span">
<span id="solution_technique2-tab8-problem4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Integrate" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_technique2-tab8-problem4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_technique2-tab8-problem4">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem4-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem4-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="technique2-tab8-problem4-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab9" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">9. Hyperbolic trig substitution</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="video" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Worked example</h3>
<div
id="video_technique2-tab9-video1"
class="video closed"
data-metadata='{"end": 0.0, "speed": null, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/publish_completion", "ytApiUrl": "https://www.youtube.com/iframe_api", "showCaptions": "true", "streams": "1.00:_kfpfMoqJgc", "poster": null, "saveStateEnabled": false, "start": 0.0, "completionPercentage": 0.95, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/transcript/available_translations", "autoplay": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/transcript/translation/__lang__", "ytMetadataEndpoint": "", "generalSpeed": 1.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "autoAdvance": false, "completionEnabled": false, "savedVideoPosition": 0.0, "saveStateUrl": "/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/xmodule_handler/save_user_state", "captionDataDir": null, "sources": [], "transcriptLanguage": "en", "recordedYoutubeIsAvailable": true, "prioritizeHls": false, "duration": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="technique2-tab9-video1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_technique2-tab9-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_technique2-tab9-video1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+18.01.2x+3T2019/xblock/block-v1:MITx+18.01.2x+3T2019+type@video+block@technique2-tab9-video1/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@vertical+block@technique2-tab10" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="vertical" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="VerticalStudentView">
<h2 class="hd hd-2 unit-title">10. Summary</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab10-text1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-usage-id="block-v1:MITx+18.01.2x+3T2019+type@html+block@technique2-tab10-text1" data-graded="True" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.01.2x+3T2019" data-block-type="html" data-request-token="3375adc8fe0b11ee8dff026cc65ec0d9" data-has-score="False" data-runtime-version="1" data-init="XBlockToXModuleShim">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Worked example with trig substitution</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Step</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Consider the integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int \frac{dx}{x^2\sqrt {1+x^2}}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Identify ugliest part of integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {1+x^2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) <table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none; border:none">
Make a substitution to simplify ugly expression:</td></tr><tr><td style="text-align:left; border:none; border:none">
Based on [mathjaxinline]\displaystyle \sec ^2\theta = 1+\tan ^2\theta[/mathjaxinline]. </td></tr></table></td><td style="text-align:left; border:none">
[mathjaxinline]\begin{cases} x= \tan \theta \\ \sqrt {1+x^2} = \sqrt {\sec ^2\theta } = \sec \theta \\ dx = \sec ^2\theta \, d\theta \end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(4) Plug into integral: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int \frac{dx}{x^2\sqrt {1+x^2}} = \int \frac{\sec \theta \, d\theta }{\tan ^2 \theta }[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(5) Rewrite in terms of [mathjaxinline]\sin[/mathjaxinline] and [mathjaxinline]\cos[/mathjaxinline]: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \int \frac{\cos \theta \, d\theta }{\sin ^2 \theta }[/mathjaxinline] </td></tr><tr><td style="text-align:left; border:none">
(6) Still not done. Need to evaluate this integral. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
How about trying a direct substitution? </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \begin{cases} u & =& \sin \theta \Longrightarrow u^2 = \sin ^2 \theta \\ du & = & \cos \theta \, d\theta \\ \end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle = \int \frac{du}{u^2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(7) Evaluate the integral. </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{-1}{u}+C[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
This is the end of the calculus. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
(8) The rest is back substitution. </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
First replace [mathjaxinline]u[/mathjaxinline] by [mathjaxinline]\sin \theta[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]= \displaystyle \frac{-1}{\sin \theta } +C[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(9) Draw a diagram of [mathjaxinline]x =\tan \theta[/mathjaxinline]: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/507e9132ced4fad75caeed675fc20201/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri1.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled x and the side adjacent to theta is labeled 1." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(10) Use diagram to determine [mathjaxinline]\displaystyle \frac{1}{\sin \theta }[/mathjaxinline] in terms of [mathjaxinline]x[/mathjaxinline]: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/26895bb377983c388dc060d1d25a3ae5/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri2.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled x and the side adjacent to theta is labeled 1. The hypotenuse is labeled square root of 1 plus x squared." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{\sin \theta } = \frac{\sqrt {1+x^2}}{x}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(11) Rewrite in terms of [mathjaxinline]x[/mathjaxinline]: </td><td style="text-align:left; border:none">
[mathjaxinline]= \displaystyle -\frac{\sqrt {1+x^2}}{x} + C[/mathjaxinline] </td></tr></table><p><b class="bfseries">Undoing trig substitution</b></p><p>
To undo a trig substitution, you must often evaluate a function of the form a trig function of an inverse trig function. For example: </p><ul class="itemize"><li><p>
[mathjaxinline]\csc \left(\arctan x \right)[/mathjaxinline] </p></li><li><p>
[mathjaxinline]\tan \left(\mathrm{arccsc} x\right)[/mathjaxinline] </p></li></ul><p>
To do this the steps are as follows: </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Undoing a trig substitution</b></td><td style="text-align:left; border:none"><b class="bf">Worked example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Compute the composite: </td><td style="text-align:left; border:none">
[mathjaxinline]\tan \underbrace{\left( \mathrm{arccsc}(x)\right)}_{\theta }[/mathjaxinline].</td></tr><tr><td style="text-align:left; border:none">
(2) Draw a picture of </td><td style="text-align:left; border:none"> </td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\begin{array}{ll}& \theta = \mathrm{arccsc}(x) \\ \Longleftrightarrow & \csc \theta = x\\ \Longleftrightarrow & \displaystyle \sin \theta = \frac{1}{x} \end{array}[/mathjaxinline] </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/89719413b0f8ed23b1033743c9c3c031/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri0.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled 1 and the hypotenuse is labeled x." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(3) Find length of remaining side: </td><td style="text-align:left; border:none"><img src="/assets/courseware/v1/8088615ba41729fdb262399606c86234/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trigsub_tri0b.svg" width="100px" alt="Right triangle with one acute angle labeled theta. The side opposite theta is labeled 1 and the hypotenuse is labeled x. The side adjacent to theta is labeled square root of x squared minus 1." style="margin: 10px 25px 25px 25px"/></td></tr><tr><td style="text-align:left; border:none">
(4) Compute trig function: </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \tan \theta = \frac{1}{\sqrt {x^2-1}}[/mathjaxinline] </td></tr></table><p>
From the pictures and formulas, we can see that we used [mathjaxinline]1 < x < \infty[/mathjaxinline] and [mathjaxinline]0 < \theta < \pi /2[/mathjaxinline]. </p><p><b class="bfseries">Summary of trig substitutions</b></p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">If integrand contains: </b></td><td style="text-align:left; border:none"><b class="bf">Make substitution:</b></td><td style="text-align:left; border:none"><b class="bf">To get:</b></td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2-x^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = \begin{cases} a\cos (\theta )\\ a\sin (\theta )\end{cases}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2-x^2}= \begin{cases} a\sin (\theta ) \\ \displaystyle a\cos \theta )\end{cases}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2+x^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = a\tan (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {a^2+x^2}= a\sec (\theta )[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {x^2-a^2}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle x = a\sec (\theta )[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {x^2-a^2}=a\tan (\theta )[/mathjaxinline] </td></tr></table><p><b class="bfseries">Completing the square</b></p><p>
We use completing the square to rewrite any quadratic function into a form which is more conducive to using trigonometric substitution. </p><p>
To complete the square means to rewrite any quadratic function [mathjaxinline]\, Ax^2+Bx+C\,[/mathjaxinline] into the form [mathjaxinline]\, A(x+b)^2+c[/mathjaxinline]. </p><p>
Here is the algebra: </p><table id="a0000002238" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002239"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle Ax^2+Bx+C[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x^2+\frac{B}{A}x \right)+C \qquad \text {Factor out }\, A[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.363)</td></tr><tr id="a0000002240"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Add and subtract the square term}\qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x^2+\frac{B}{A}x+ \left(\frac{B}{2A} \right)^2 \right) - A\left(\frac{B}{2A} \right)^2+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.364)</td></tr><tr id="a0000002241"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle A\left(x+\frac{B}{2A} \right)^2 +\left(-\frac{B^2}{4A}+C\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.365)</td></tr></table>
</div>
</div>
</div>
</div>
© All Rights Reserved