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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Integration by parts</h3>
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<h2 class="hd hd-2 unit-title">2. Integration by parts</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Use <span style="color:#27408C"><b class="bf">integration by parts</b></span> to evaluate integrals.<br/></p></li><li><p>
Derive and use <span style="color:#27408C"><b class="bf">reduction formulas</b></span>. </p></li></ul><p><b class="bfseries">Contents: 13 pages</b></p><p>
6 videos (42 minutes 1x speed) 20 questions </p>
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<h2 class="hd hd-2 unit-title">3. Integration by parts</h2>
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Review: integral of products
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True or false: [mathjaxinline]\displaystyle \int x\cdot \frac{1}{x}\, dx\, =\, \left(\int x\, dx\right)\cdot \left(\int \frac{1}{x}\, dx\right)\,[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_technique4-tab3-problem1_2_1">
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True or False: [mathjaxinline]\displaystyle \int u(x) \cdot v(x)\, dx\, =\, \left(\int u(x) \, dx\right)\cdot \left(\int v(x)\, dx\right)\,[/mathjaxinline]? </p>
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<p><b class="bfseries">Review: the differential</b></p><p>
Recall that the differential [mathjaxinline]\, du\,[/mathjaxinline] is defined as </p><table id="a0000002414" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002415"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \, du[/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 u'(x)\, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.430)</td></tr></table><p>
The abbreviation [mathjaxinline]\, du[/mathjaxinline] is very convenient as you will see below.<br/></p>
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Integrating the product rule
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Find [mathjaxinline]\, d(u\, v)[/mathjaxinline] in terms of [mathjaxinline]\, u,\,[/mathjaxinline] [mathjaxinline]\, du,\,[/mathjaxinline] [mathjaxinline]\, v,\,[/mathjaxinline] [mathjaxinline]\, dv[/mathjaxinline]. </p>
<p>
(Type &#8220;du" for the differential [mathjaxinline]\, du[/mathjaxinline]. Your answer should <b class="bf">not</b> contain [mathjaxinline]\, dx[/mathjaxinline]. Please use explicit multiplication between functions and differentials: i.e. u*du, NOT udu.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle d \left(u\, v\right) \, =\,[/mathjaxinline]</p>
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Integrate the product rule above. Choose all formulations below that correctly describe what we get. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_technique4-tab3-problem2_3_1">
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<input type="checkbox" name="input_technique4-tab3-problem2_3_1[]" id="input_technique4-tab3-problem2_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="technique4-tab3-problem2_3_1-choice_0-label" for="input_technique4-tab3-problem2_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_technique4-tab3-problem2_3_1"> <text>[mathjaxinline]\displaystyle u\, v\, +C\, =\, \int \left(u(x)'\, v(x)'\right)\, dx[/mathjaxinline]</text>
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<input type="checkbox" name="input_technique4-tab3-problem2_3_1[]" id="input_technique4-tab3-problem2_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="technique4-tab3-problem2_3_1-choice_1-label" for="input_technique4-tab3-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_technique4-tab3-problem2_3_1"> <text>[mathjaxinline]\displaystyle u\, v\, +C\, =\, \int \left(du \, dv\right)[/mathjaxinline]</text>
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<input type="checkbox" name="input_technique4-tab3-problem2_3_1[]" id="input_technique4-tab3-problem2_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="technique4-tab3-problem2_3_1-choice_2-label" for="input_technique4-tab3-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_technique4-tab3-problem2_3_1"> <text>[mathjaxinline]\displaystyle u\, v\, +C\, =\, \int v\, du+\int u\, dv[/mathjaxinline]</text>
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<input type="checkbox" name="input_technique4-tab3-problem2_3_1[]" id="input_technique4-tab3-problem2_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="technique4-tab3-problem2_3_1-choice_3-label" for="input_technique4-tab3-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_technique4-tab3-problem2_3_1"> <text>[mathjaxinline]\displaystyle u\, v\, +C\, =\, \int \left( v\, \frac{du}{dx}\right) \, dx+\int \left(u\, \frac{dv}{dx}\right)\, dx[/mathjaxinline]</text>
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<h3 class="hd hd-2">Integration by parts and first example</h3>
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<h2 class="hd hd-2 unit-title">4. Integration by parts</h2>
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<p><span style="color:#27408C"><b class="bf">Integration by parts</b></span> is the integral version of the product rule for differentiation. <br/></p><table id="a0000002423" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002424"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int u\, v' \, dx[/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 u\, v-\int u'\, v\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.436)</td></tr><tr id="a0000002425"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _ a^ b u\, v' \, dx[/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 \left.\phantom{\int } u\, v\, \right|_ a^ b-\int _ a^ b u'\, v\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.437)</td></tr></table><p>
The second formula, for the definite integral, is obtained by using FTC1 on the antiderivative. We will assume you know how to find a definite integral from the antiderivative, and only practice finding indefinite integrals using integration by parts in this section.<br/></p><p><b class="bfseries">Example: Integral of the logarithm</b></p><p>
We are presenting three versions of integration by parts using slightly different notation. You can use whichever one you prefer.<br/></p><p><b class="bfseries">Version 1</b></p><table id="a0000002426" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002427"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \ln (x)\, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'} \, dx[/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 \int \, u\, v'\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.438)</td></tr></table><table id="a0000002428" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002429"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u\, =\, \ln (x)[/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 v'\, =\, 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.439)</td></tr><tr id="a0000002430"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, \frac{1}{x}[/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 v\, =\, x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.440)</td></tr></table><table id="a0000002431" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002432"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'}\, dx[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} \, -\, \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.441)</td></tr><tr id="a0000002433"><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 x\ln (x)\, -\, x\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.442)</td></tr></table><p><b class="bfseries">Version 2</b></p><table id="a0000002434" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002435"><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 u\, v' dx[/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 \ln (x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.443)</td></tr><tr id="a0000002436"><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 u\, =\, \ln (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle v'\, =\, 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.444)</td></tr><tr id="a0000002437"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, \frac{1}{x}[/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 v=x.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.445)</td></tr></table><table id="a0000002438" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002439"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'} \, dx[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} \, -\, \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.446)</td></tr><tr id="a0000002440"><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 x\ln (x)\, -\, x\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.447)</td></tr></table><p><b class="bfseries">Version 3</b></p><table id="a0000002441" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002442"><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 u\, dv[/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 \ln (x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.448)</td></tr><tr id="a0000002443"><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 u\, =\, \ln (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle dv\, =\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.449)</td></tr><tr id="a0000002444"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle du\, =\, \frac{1}{x}\, dx[/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 v=x.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.450)</td></tr></table><table id="a0000002445" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002446"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{dx}_{dv}[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} - \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}\, dx}_{du}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.451)</td></tr><tr id="a0000002447"><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 x\ln (x)- x+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.452)</td></tr></table><p><b class="bf">Note:</b> When [mathjaxinline]v'=1,[/mathjaxinline] we chose [mathjaxinline]\, v=x\,[/mathjaxinline] not [mathjaxinline]\, v=x+23\,[/mathjaxinline] or [mathjaxinline]\, v=x+C[/mathjaxinline]. This is because every choice of antiderivative works, so we choose the simplest one.<br/></p><p>
A key issue in the procedure is to keep track of the data: [mathjaxinline]\, u,\,[/mathjaxinline] [mathjaxinline]\, u',\,[/mathjaxinline] [mathjaxinline]\, v,\,[/mathjaxinline] and [mathjaxinline]\, v'\,[/mathjaxinline].<br/></p><p>
Once [mathjaxinline]u[/mathjaxinline] is chosen, the procedure is mechanical and determined. In this example, we chose [mathjaxinline]\, u=\ln (x).\, \,[/mathjaxinline] How do we choose [mathjaxinline]\, u\,[/mathjaxinline] in general? The goal is to choose [mathjaxinline]\, u\,[/mathjaxinline] so that we can replace a harder integral with an easier one. <br/></p><p>
As you practice more, you will learn when to use integration by parts and what [mathjaxinline]\, u[/mathjaxinline] to choose. Let us start by practicing using the formula.<br/></p>
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Practice 1
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<p>
Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int \arctan (x)\, dx[/mathjaxinline].<br/></p>
<p>
Set [mathjaxinline]\, u=\arctan (x)\,[/mathjaxinline] and [mathjaxinline]\, v'=1,\,[/mathjaxinline] so that [mathjaxinline]\, u\, v'\, =\, \arctan (x),\,[/mathjaxinline] the integrand. Find [mathjaxinline]\, u'\,[/mathjaxinline] and [mathjaxinline]\, v[/mathjaxinline].<br/>(Choose the simplest [mathjaxinline]\, \displaystyle v \,[/mathjaxinline].)<br/><p style="display:inline"> [mathjaxinline]\, u'\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique4-tab4-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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</div></div><br/><p style="display:inline">[mathjaxinline]\, v\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique4-tab4-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
Write the formula for integration by parts:<br/></p>
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<p style="text-align:left"> \(\displaystyle \large{\int \arctan(x)\,dx} = \)</p>
</td>
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<br/>
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<p style="display:inline; text-align:left"> \( \displaystyle \large{- \int } \)</p>
</td>
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<br/>
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<span class="trailing_text" id="trailing_text_technique4-tab4-problem1_5_1"> [mathjaxinline] \large{dx}. [/mathjaxinline]</span>
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<p>
Evaluate the integral. </p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
<p>
<p style="display:inline">[mathjaxinline]\, \displaystyle {\large \int \arctan (x)\, dx\, =\, }[/mathjaxinline]</p>
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<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>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<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 \(\lambda \)
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<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 \( 2\pi \)
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<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 \( \sqrt{x^2-y} \)
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<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>
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<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 \( (2\pi+y)dy \)
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Practice 2
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Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int \arcsin (x)\, dx[/mathjaxinline].<br/></p>
<p>
(Hint: Choose [mathjaxinline]\, u\,[/mathjaxinline] and [mathjaxinline]v'\,[/mathjaxinline] in a similar way to the previous problem.)<br/></p>
<p>
<p style="display:inline"> Set [mathjaxinline]\, u\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\, v'\, =\,[/mathjaxinline]</p>
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<p style="display:inline"> [mathjaxinline]\, u'\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\, v\, =\,[/mathjaxinline]</p>
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\(\)
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<br/>
</p>
<p>
Write the formula for integration by parts:<br/><span><style>
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.xmodule_display.xmodule_CapaModule div.problem section div span.MathJax {
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</style><table><tbody><tr><td><p style="text-align:left"> \(\displaystyle \large{\int \arcsin(x)\,dx} = \)</p></td><td style="padding-top: 28px"><br/><div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="formulaequationinput_technique4-tab4-problem2_6_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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\(\)
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td><td><p style="display:inline; text-align:left"> \( \displaystyle \large{- \int } \)</p></td><td style="padding-top: 28px"><br/><div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="formulaequationinput_technique4-tab4-problem2_7_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique4-tab4-problem2_7_1" id="input_technique4-tab4-problem2_7_1" data-input-id="technique4-tab4-problem2_7_1" value="" aria-describedby="trailing_text_technique4-tab4-problem2_7_1 status_technique4-tab4-problem2_7_1" size="15"/>
<span class="trailing_text" id="trailing_text_technique4-tab4-problem2_7_1"> [mathjaxinline] \large{dx.} [/mathjaxinline]</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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\(\)
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</div>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div></td></tr></tbody></table></span></p>
<p>
Evaluate the integral. </p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
<p>
<p style="display:inline">[mathjaxinline]\, \displaystyle {\large \int \arcsin (x)\, dx\, }=\,[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
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<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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>
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<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>
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<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>
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<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>
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Practice 3
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<p>
We will use Integration by parts to evaluate [mathjaxinline]\, \displaystyle \int x \, \ln (x)\, dx[/mathjaxinline].<br/></p>
<p>
Set [mathjaxinline]\, u=\ln (x)\,[/mathjaxinline] and [mathjaxinline]\, v'=x,\,[/mathjaxinline] so that [mathjaxinline]\, u v'\, =\, x\, \ln (x),\,[/mathjaxinline] the integrand. <br/></p>
<p>
Write the formula for integration by parts:<br/></p>
<span>
<style>
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min-width: 0 !important;
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<td>
<p style="text-align:left"> \(\displaystyle \large{\int x \ln(x)\, dx} = \)</p>
</td>
<td style="padding-top: 28px">
<br/>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique4-tab5-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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</td>
<td>
<p style="display:inline; text-align:right"> \( \displaystyle \large{- \int }\)</p>
</td>
<td style="padding-top: 28px">
<br/>
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<span class="trailing_text" id="trailing_text_technique4-tab5-problem1_3_1"> [mathjaxinline] \large{dx}. [/mathjaxinline]</span>
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<p>
Evaluate the integral. </p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
<p>
<p style="display:inline">[mathjaxinline]\, \displaystyle {\large \int x\, \ln (x)\, dx\, =\, }[/mathjaxinline]</p>
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<tr class="fiptitle">
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<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>
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<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>
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<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>
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<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>
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<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 \( (2\pi+y)dy \)
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Practice 4
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We can modify our computation in the previous problem to evaluate a more general integral.<br/></p>
<p>
Let [mathjaxinline]\, p\neq -1[/mathjaxinline]. Integrate [mathjaxinline]\, \displaystyle \int x^ p \, \ln (x)\, dx\,[/mathjaxinline] by parts.<br/></p>
<p>
As in the previous problem, set [mathjaxinline]\, u=\ln (x)\,[/mathjaxinline]. Find [mathjaxinline]\, v',[/mathjaxinline] [mathjaxinline]\, u',\,[/mathjaxinline] [mathjaxinline]\, v[/mathjaxinline] and carry out the integration by parts.<br/></p>
<p>
Write the formula for integration by parts:<br/></p>
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<p style="text-align:left"> \(\displaystyle \large{\int x^p\,\ln(x)\,dx} = \)</p>
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<br/>
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\(\)
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<p style="display:inline; text-align:right"> \( \displaystyle \large{- \int }\)</p>
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<span class="trailing_text" id="trailing_text_technique4-tab5-problem2_3_1"> [mathjaxinline] \large{dx}. [/mathjaxinline]</span>
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<p>
Evaluate the integral. </p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
<p>
<p style="display:inline">[mathjaxinline]\, \displaystyle {\large \int x^ p\, \ln (x)\, dx\, =\, }[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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>
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<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>
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<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>
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<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 \( (2\pi+y)dy \)
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Choosing u 1
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<p>
Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int x \, \sin (x)\, dx[/mathjaxinline].<br/></p>
<p>
There are two natural choices of [mathjaxinline]u[/mathjaxinline] ( which determines [mathjaxinline]v'[/mathjaxinline]).<br/></p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002448" style="table-layout:auto" width="100%">
<tr id="a0000002449">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u_1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =\, x,[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v_1'[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\sin (x) \qquad \text {OR}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.453)</td>
</tr>
<tr id="a0000002450">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u_2[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =\sin (x),[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v_2'[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =x[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.454)</td>
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Find [mathjaxinline]\, u_1',\,[/mathjaxinline] [mathjaxinline]\, v_1,\,[/mathjaxinline] and [mathjaxinline]\, u_2',\,[/mathjaxinline][mathjaxinline]\, v_2[/mathjaxinline].<br/><p style="display:inline"> [mathjaxinline]\, u_1'\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique4-tab6-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Which choice do you prefer? Use it to compute the integral. </p>
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(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\, \displaystyle \int x\, \sin (x)\, dx\, =\,[/mathjaxinline]</p>
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Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int x \, e^{2x}\, dx[/mathjaxinline]. </p>
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(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\, \displaystyle \int x\, e^{2x}\, dx\, =\,[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<td class="formulainput">Integers</td>
<td class="formulainput">
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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 \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">7. Worked examples</h2>
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Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int x^3 \, \cos (x^2)\, dx[/mathjaxinline].<br/></p><p>
We choose </p><table id="a0000002459" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002460"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u\, =\, x^2[/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 v'\, =\, x\, \cos \left(x^2\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.461)</td></tr><tr id="a0000002461"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, 2x[/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 v\, =\, \frac{\sin \left(x^2\right)}{2}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.462)</td></tr></table><p>
Integrating by parts, we have </p><table id="a0000002462" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002463"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \underbrace{x^2}_ u \, \, \underbrace{x\cos \left(x^2\right)}_{v'}\, dx\, \,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \underbrace{x^2}_{u}\, \, \underbrace{\frac{\sin \left(x^2\right)}{2}}_{v} - \int \underbrace{\frac{\sin \left(x^2\right)}{2}}_{v}\, \, \underbrace{2x}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.463)</td></tr><tr id="a0000002464"><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:left; border:none">
[mathjaxinline]\displaystyle = \frac{x^2 \sin \left(x^2\right)}{2}+ \frac{\cos \left(x^2\right)}{2}+C.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.464)</td></tr></table><p>
How did we choose [mathjaxinline]\, u\,[/mathjaxinline] and [mathjaxinline]\, v'\,[/mathjaxinline]? The short answer is that we chose the one that worked! <br/></p><p>
The longer answer is as follows. First, look at the most complicated function [mathjaxinline]\, \cos \left(x^2\right),\,[/mathjaxinline] in the integrand, if we set [mathjaxinline]\, u\, =\, \cos \left(x^2\right),\,[/mathjaxinline] then </p><table id="a0000002465" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002466"><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 u\, =\, \cos \left(x^2\right)[/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 v\, =\, x^3[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.465)</td></tr><tr id="a0000002467"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, - 2x\, \sin \left(x^2\right)[/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 v\, =\, \frac{x^4}{4}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.466)</td></tr></table><p>
The derivative of [mathjaxinline]\, u,\,[/mathjaxinline] is complicated and we check that [mathjaxinline]\, \displaystyle \int u' v \, dx\,[/mathjaxinline] is also more complicated than the original integral: </p><table id="a0000002468" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002469"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int u'v\, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int 2x\, \sin \left(x^2\right)\, \left(\frac{x^4}{4}\right)\, dx[/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 \frac{1}{2}\, \int x^5\, \sin \left(x^2\right)\, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.467)</td></tr></table><p>
Next, we try [mathjaxinline]\, v'=\cos \left(x^2\right),\, \,[/mathjaxinline] but we cannot integrate it easily. However, we do know how to integrate [mathjaxinline]\, x\cos \left(x^2\right).\, \,[/mathjaxinline] Therefore, the trick is to choose </p><table id="a0000002470" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002471"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle v'\, =\, x\cos \left(x^2\right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle v\, =\, \frac{1}{2} \sin \left(x^2\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.468)</td></tr></table><p>
Once this choice is made, the rest of the procedure is mechanical and as above.<br/></p><p><b class="bf">Note:</b> Recall [mathjaxinline]\, \displaystyle \int \cos (x^2)\, dx[/mathjaxinline] is the Fresnel integral and cannot be expressed in terms of elementary functions. </p>
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<p>
Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int x^3 \sqrt {x^2+7}\, dx[/mathjaxinline].<br/></p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\, \displaystyle \int x^3 \sqrt {x^2+7}\, dx\, =\,[/mathjaxinline]</p>
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<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>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
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<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>
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<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>
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<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 \( 2\pi \)
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<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>
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<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>
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<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/>
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Practice 2
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Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int \frac{x^5}{\left(x^3+1\right)^5}\, dx[/mathjaxinline].<br/></p>
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(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\, \displaystyle \int \frac{x^5}{\left(x^3+1\right)^5}\, dx\, =\,[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<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 \(\lambda \)
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<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 \( 2\pi \)
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<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 \( \sqrt{x^2-y} \)
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<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>
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<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 \( (2\pi+y)dy \)
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Practice 3
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Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int \sin (2x) \, e^{\cos (x)} \, dx[/mathjaxinline].<br/></p>
<p><i class="itshape">Hint</i>: First use the double angle formula for [mathjaxinline]\, \sin (2x)\,[/mathjaxinline].<br/></p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] for the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\, \displaystyle \int \sin (2x) \, e^{\cos (x)}\, dx\, =\,[/mathjaxinline]</p>
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<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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 \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">8. Examples with reduction formula</h2>
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Another reduction formula of the logarithm
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<p>
In a previous problem, we have evaluated [mathjaxinline]\, \displaystyle \int x^ p (\ln \, x\, )\, dx\,[/mathjaxinline] for [mathjaxinline]\, p\neq -1[/mathjaxinline].<br/></p>
<p>
We will now find a reduction formula for [mathjaxinline]\, \displaystyle \int x^ p (\ln \, x\, )^ n \, dx\,[/mathjaxinline] where [mathjaxinline]\, n \geq 1\,[/mathjaxinline] is an integer, and [mathjaxinline]\, p\neq - 1[/mathjaxinline] as before. Note that [mathjaxinline]\, p[/mathjaxinline] remains the same in [mathjaxinline]\, I_ n[/mathjaxinline] for any [mathjaxinline]\, n[/mathjaxinline].<br/></p>
<p>
Define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002498" style="table-layout:auto" width="100%">
<tr id="a0000002499">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle I_ n[/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 \int x^ p (\ln \, x\, )^ n \, dx \qquad (\text {for fixed}\, \, p\neq -1)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.486)</td>
</tr>
</table>
<p>
Use integration by parts to write [mathjaxinline]\, I_ n\,[/mathjaxinline] in terms of [mathjaxinline]\, I_{n-1}\,[/mathjaxinline].<br/>(Your answers should be in terms of [mathjaxinline]\, p,\, x,\,[/mathjaxinline] and [mathjaxinline]\, n[/mathjaxinline].)<br/></p>
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<p style="text-align:left"> \(\displaystyle \Large{I_n \,=\,} \)</p>
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<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
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Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Using the reduction formula
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As above, define </p>
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[mathjaxinline]\displaystyle \displaystyle I_ n[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int x^ p (\ln \, x\, )^ n \, dx \qquad (\text {for fixed}\, \, p\neq 1).[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.492)</td>
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Use this definition to find [mathjaxinline]I_0[/mathjaxinline], and use the reduction formula in the previous problem to evaluate [mathjaxinline]\, I_1,\,[/mathjaxinline] and [mathjaxinline]\, I_2[/mathjaxinline].<br/></p>
<p>
(Use [mathjaxinline]\, C\,[/mathjaxinline] for the constant of integration.)<br/>(Your answers should be in terms of [mathjaxinline]\, p,\, x[/mathjaxinline] and the constant of integration [mathjaxinline]\, C[/mathjaxinline]. )<br/></p>
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<p style="display:inline">[mathjaxinline]\, I_0\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\, I_1\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\, I_2\, =\,[/mathjaxinline]</p>
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Reduction formula: powers of x times sine
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Find a reduction formula for [mathjaxinline]\, \displaystyle \int x^ n \sin (x) \, dx\,[/mathjaxinline] for any integer [mathjaxinline]\, n \geq 0[/mathjaxinline]. </p>
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Define </p>
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[mathjaxinline]\displaystyle \displaystyle S_ n[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int x^ n \sin (x) \, dx \qquad (n\geq 0 \, \, \text {integer}).[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.501)</td>
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Write [mathjaxinline]\, S_ n\,[/mathjaxinline] in terms of [mathjaxinline]\, S_{n-2}\,[/mathjaxinline] by integrating by parts <b class="bf">twice</b>. </p>
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<p style="text-align:left"> \(\displaystyle \Large{S_n \,=\,} \)</p>
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<span class="trailing_text" id="trailing_text_technique4-tab9-problem1_3_1">[mathjaxinline]\displaystyle \Large{S_{n-2}} [/mathjaxinline]</span>
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<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<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 \(\lambda \)
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<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 \( 2\pi \)
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<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 \( \sqrt{x^2-y} \)
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<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>
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<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>
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<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>
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<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 \( (2\pi+y)dy \)
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Using the reduction formula
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As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002537" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle S_ n[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int x^ n \sin (x) \, dx \qquad (n\geq 0 \, \, \text {integer}).[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.511)</td>
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Use the reduction formula in the previous problem to find [mathjaxinline]\, \displaystyle \int _0^{\pi } x^2 \sin (x)\, dx[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\, \displaystyle \int _0^\pi x^2 \sin (x)\, dx=\,[/mathjaxinline]</p>
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<font color="#0078b0">2/3</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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 \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">10. Worked example: reappearance of the original integral</h2>
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<p>
Let us look at an example in which the original integral reappears on the right hand side of the formula after applying integration by parts.<br/></p><p>
Use integration by parts to evaluate [mathjaxinline]\, \displaystyle \int e^ x \cos (x)\, dx[/mathjaxinline].<br/></p><table id="a0000002549" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002550"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =e^ x;[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\cos (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.518)</td></tr><tr id="a0000002551"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =e^ x;[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\sin (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.519)</td></tr></table><table id="a0000002552" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002553"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \underbrace{e^ x}_ u \, \, \underbrace{\cos (x)}_{v'}\, dx\, \,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad \underbrace{e^ x}_{u}\, \, \underbrace{\sin (x)}_{v} - \int \underbrace{\sin (x)}_{v}\, \, \underbrace{e^ x}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.520)</td></tr><tr id="a0000002554"><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:left; border:none">
[mathjaxinline]\displaystyle = e^ x\sin (x) - \, \int e^ x\, \sin (x)\, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.521)</td></tr></table><p>
Next, we integrate by parts again to try to evaluate the resulting integral [mathjaxinline]\, \displaystyle \int e^ x\, \sin (x)\, dx\,[/mathjaxinline]: </p><table id="a0000002555" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002556"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =e^ x;[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\sin (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.522)</td></tr><tr id="a0000002557"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u'[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =e^ x;[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle v[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =-\cos (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.523)</td></tr></table><table id="a0000002558" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002559"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \underbrace{e^ x}_ u \, \, \underbrace{\sin (x)}_{v'}\, dx\, \,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \quad \underbrace{e^ x}_{u}\, \, \underbrace{\left(-\cos (x)\right)}_{v} - \int \underbrace{\left(-\cos (x)\right)}_{v}\, \, \underbrace{e^ x}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.524)</td></tr><tr id="a0000002560"><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:left; border:none">
[mathjaxinline]\displaystyle = -e^ x\cos (x) + \, \int e^ x\, \cos (x)\, dx.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.525)</td></tr></table><p>
Therefore, combining the above, we get : </p><table id="a0000002561" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002562"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int e^ x \, \cos (x)\, dx\, \,[/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 e^ x\sin (x) \, - \, \int e^ x\, \sin (x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.526)</td></tr><tr id="a0000002563"><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 e^ x\sin (x)\, -\, \left(-e^ x\cos (x) + \, \int e^ x\, \cos (x)\, dx\, \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.527)</td></tr><tr id="a0000002564"><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 e^ x\, \left(\sin (x)\, +\, \cos (x)\right) - \, \int e^ x\, \cos (x)\, dx.\,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.528)</td></tr></table><p>
The integral we started with [mathjaxinline]\, \displaystyle \int e^ x\, \cos (x)\, dx\,[/mathjaxinline] appears again on the right hand side of our formula! It looks like we have just gone around in circles, but actually we are almost done, because the unknown integral appears with coefficient [mathjaxinline]+1[/mathjaxinline] on the left hand side and [mathjaxinline]-1[/mathjaxinline] on the right hand side, and we so can solve for it by adding it to both sides:<br/></p><table id="a0000002565" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002566"><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 \int e^ x \, \cos (x)\, dx[/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 e^ x\, \left(\sin (x)\, +\, \cos (x)\right) - \, \int e^ x\, \cos (x)\, dx.\,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.529)</td></tr><tr id="a0000002567"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 2\, \left( \int e^ x \, \cos (x)\, dx \right)[/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 = e^ x\, \left(\sin (x)\, +\, \cos (x)\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.530)</td></tr><tr id="a0000002568"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int e^ x\, \cos (x)\, dx[/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 \frac{e^ x \left(\sin (x)+\cos (x)\right)}{2}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.531)</td></tr></table>
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<h2 class="hd hd-2 unit-title">11. Trig reduction formulas</h2>
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Integral of powers of cosine
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When we discussed integrals of trigonometric functions, we did not have formulas for the integral of even positive powers of [mathjaxinline]\, \sin \,[/mathjaxinline] or [mathjaxinline]\, \cos \,[/mathjaxinline]. We will now find reduction formulas for them.<br/></p>
<p>
Define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002569" style="table-layout:auto" width="100%">
<tr id="a0000002570">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle C_ n[/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 \int \cos ^ n(x)\, dx.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.532)</td>
</tr>
</table>
<p>
Using the trigonometric identity [mathjaxinline]\, \displaystyle \sin ^2(x)+\cos ^2(x)=1,\,[/mathjaxinline] we get </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002571" style="table-layout:auto" width="100%">
<tr id="a0000002572">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle C_ n[/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 C_{n-2} - \int \cos ^{n-2}(x)\sin ^2(x)\, dx[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.533)</td>
</tr>
</table>
<p>
Use integration by parts to write [mathjaxinline]\, \displaystyle \int \cos ^{n-2}(x)\sin ^2(x)\, dx\,[/mathjaxinline] in terms of [mathjaxinline]\, C_ n\,[/mathjaxinline] where [mathjaxinline]\, n\neq 1.\, \,[/mathjaxinline] Write the resulting reduction formula for [mathjaxinline]\, C_ n,\,[/mathjaxinline] with [mathjaxinline]\, n\neq 1\,[/mathjaxinline]below.<br/></p>
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<p style="text-align:left"> \(\displaystyle \Large{C_n \,=\,} \)</p>
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<span class="trailing_text" id="trailing_text_technique4-tab11-problem1_2_1">[mathjaxinline]\displaystyle \Large{+}[/mathjaxinline]</span>
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<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique4-tab11-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_technique4-tab11-problem1_3_1">[mathjaxinline]\displaystyle \Large{C_{n-2}}[/mathjaxinline]</span>
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Integral of cosine squared
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<p>
As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002585" style="table-layout:auto" width="100%">
<tr id="a0000002586">
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[mathjaxinline]\displaystyle \displaystyle C_ n[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int \cos ^ n(x)\, dx \qquad \text {for} \, n\geq 0.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.543)</td>
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<p>
Find [mathjaxinline]\, C_0,\,[/mathjaxinline] and use the reduction formula in the previous problem to evaluate [mathjaxinline]\, C_2[/mathjaxinline].<br/></p>
<p>
(Use [mathjaxinline]\, C\,[/mathjaxinline] for the constant of integration.)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\, C_{0}\, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\, C_{2}=\,[/mathjaxinline]</p>
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Integral of powers of secant(*)
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<p>
When we discussed integrals of trigonometric functions, we did not have formulas for the integral of odd positive powers of [mathjaxinline]\, \sec \,[/mathjaxinline]. <br/></p>
<p>
We can find a reduction formula for it directly using integration by parts, but we can also derive it from the reduction formula for the integral of powers of cosine.<br/></p>
<p>
As above,define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002594" style="table-layout:auto" width="100%">
<tr id="a0000002595">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle C_ n[/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 \int \cos ^ n(x)\, dx.[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.548)</td>
</tr>
</table>
<p>
Now, also define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002596" style="table-layout:auto" width="100%">
<tr id="a0000002597">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \zeta _ m[/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 \int \sec ^ m(x)\, dx.[/mathjaxinline]
</td>
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.549)</td>
</tr>
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<p>
Find [mathjaxinline]\, m\,[/mathjaxinline] such that [mathjaxinline]\, \zeta _ m\, =\, C_ n[/mathjaxinline]. (Enter [mathjaxinline]\, m\,[/mathjaxinline] in terms of [mathjaxinline]\, n[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]\, m =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique4-tab11-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Check that the reduction formula for [mathjaxinline]\, C_ n\,[/mathjaxinline] is valid for [mathjaxinline]\, n&lt; 0\,[/mathjaxinline] and rearrange it to obtain a reduction formula for [mathjaxinline]\, \zeta _ m\,[/mathjaxinline] where [mathjaxinline]\, m\neq 1,2.\, \,[/mathjaxinline] Enter your formula below.<br/></p>
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<p style="text-align:left"> \(\displaystyle \Large{\zeta_{m} \,=\,} \)</p>
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<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_technique4-tab11-problem3_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<span class="trailing_text" id="trailing_text_technique4-tab11-problem3_4_1">[mathjaxinline]\displaystyle \Large{\zeta_{m-2}} [/mathjaxinline]</span>
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<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<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>
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<td class="formulainput">e, pi</td>
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Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<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>
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<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<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>
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<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 \( (2\pi+y)dy \)
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Integral of secant cubed
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As above, define </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002613" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \zeta _ m[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int \sec ^ m(x)\, dx.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.559)</td>
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Use [mathjaxinline]\, \displaystyle \zeta _1= \int \sec (x)\, dx = \ln \left|\sec (x)+\tan (x)\right| +C \,[/mathjaxinline] and the reduction formula for [mathjaxinline]\, \zeta _ m\,[/mathjaxinline] from the previous problem to evaluate </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000002615" style="table-layout:auto" width="100%">
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \zeta _3= \int \sec ^3(x)\, dx.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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(Use [mathjaxinline]\, C\,[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\, \zeta _{3}\, =\,[/mathjaxinline]</p>
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<font color="#0078b0">2/3</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<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>
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<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 \( 2\pi \)
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<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>
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<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 \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">13. Summary</h2>
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<p><b class="bfseries">Integration by parts and a first example</b></p><p><span style="color:#27408C"><b class="bf">Integration by parts</b></span> is the integral version of the product rule for differentiation. <br/></p><table id="a0000002623" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002624"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int u\, v' \, dx[/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 u\, v-\int u'\, v\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.564)</td></tr><tr id="a0000002625"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int _ a^ b u\, v' \, dx[/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 \left.\phantom{\int } u\, v\, \right|_ a^ b-\int _ a^ b u'\, v\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.565)</td></tr></table><p><b class="bfseries">Example: Integral of the logarithm</b></p><p>
We are presenting three versions of integration by parts using slightly different notation. You can use whichever one you prefer.<br/></p><p><b class="bfseries">Version 1</b></p><table id="a0000002626" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002627"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \ln (x)\, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'} \, dx[/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 \int \, u\, v'\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.566)</td></tr></table><table id="a0000002628" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002629"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u\, =\, \ln (x)[/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 v'\, =\, 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.567)</td></tr><tr id="a0000002630"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, \frac{1}{x}[/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 v\, =\, x[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.568)</td></tr></table><table id="a0000002631" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002632"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'}\, dx[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} \, -\, \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.569)</td></tr><tr id="a0000002633"><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 x\ln (x)\, -\, x\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.570)</td></tr></table><p><b class="bfseries">Version 2</b></p><table id="a0000002634" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002635"><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 u\, v' dx[/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 \ln (x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.571)</td></tr><tr id="a0000002636"><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 u\, =\, \ln (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle v'\, =\, 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.572)</td></tr><tr id="a0000002637"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle u'\, =\, \frac{1}{x}\, dx[/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 v=x.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.573)</td></tr></table><table id="a0000002638" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002639"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{1}_{v'} \, dx[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} \, -\, \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}}_{u'}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.574)</td></tr><tr id="a0000002640"><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 x\ln (x)\, -\, x\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.575)</td></tr></table><p><b class="bfseries">Version 3</b></p><table id="a0000002641" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002642"><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 u\, dv[/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 \ln (x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.576)</td></tr><tr id="a0000002643"><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 u\, =\, \ln (x)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \Rightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle dv\, =\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.577)</td></tr><tr id="a0000002644"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle du\, =\, \frac{1}{x}\, dx[/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 v=x.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.578)</td></tr></table><table id="a0000002645" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002646"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \underbrace{\ln (x)}_ u \, \, \underbrace{dx}_{dv}[/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 \underbrace{x}_{v}\, \, \underbrace{\ln (x)}_{u} - \int \underbrace{x}_{v}\, \, \underbrace{\frac{1}{x}\, dx}_{du}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.579)</td></tr><tr id="a0000002647"><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 x\ln (x)- x+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.580)</td></tr></table><p><b class="bf">Note:</b> When [mathjaxinline]v'=1,[/mathjaxinline] we chose [mathjaxinline]\, v=x\,[/mathjaxinline] not [mathjaxinline]\, v=x+23\,[/mathjaxinline] or [mathjaxinline]\, v=x+C[/mathjaxinline]. This is because every choice of antiderivative works, so we choose the simplest one.<br/></p><p>
A key issue in the procedure is to keep track of the data: [mathjaxinline]\, u,\,[/mathjaxinline] [mathjaxinline]\, u',\,[/mathjaxinline] [mathjaxinline]\, v,\,[/mathjaxinline], [mathjaxinline]\, v'\,[/mathjaxinline].<br/></p><p>
Once [mathjaxinline]u[/mathjaxinline] is chosen, the procedure is mechanical and determined. In this example, we chose [mathjaxinline]\, u=\ln (x).\, \,[/mathjaxinline] How do we choose [mathjaxinline]\, u\,[/mathjaxinline] in general? The goal is to choose [mathjaxinline]\, u\,[/mathjaxinline] so that we can replace a harder integral with an easier one. <br/></p>
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