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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Trig integrals</h3>
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<h2 class="hd hd-2 unit-title">2. Integrals of trigonometric functions</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Use trigonometric identities and the method of substitution to evaluate [mathjaxinline]\displaystyle \displaystyle \int \sin ^ m(\theta ) \cos ^ n(\theta ) \, d\theta[/mathjaxinline] in the following three cases: </p><ol class="enumerate"><li value="1"><p>
when either [mathjaxinline]\, m\,[/mathjaxinline] or [mathjaxinline]\, n\,[/mathjaxinline] is an odd positive integer, </p></li><li value="2"><p>
when both [mathjaxinline]\, m\,[/mathjaxinline] and [mathjaxinline]\, n\,[/mathjaxinline] are even positive integers, </p></li><li value="3"><p>
when [mathjaxinline]\, m+n\,[/mathjaxinline] is an even negative integer. </p></li></ol></li></ul><p><b class="bfseries">Contents: 16 pages</b></p><p>
10 videos (56 minutes 1x speed) 41 questions </p>
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<h2 class="hd hd-2 unit-title">3. Summary of formulas</h2>
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<p><b class="bfseries">Trigonometric identities</b></p><p>
We will be using the following trigonometric identities throughout this section.<br/></p><p><b class="bfseries">The Pythagorean theorem</b></p><table id="a0000001714" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001715"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos ^2(\theta )+\sin ^2(\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.1)</td></tr><tr id="a0000001716"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle 1+\tan ^2(\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sec ^2(\theta )\qquad \left(\theta \neq \frac{\pi }{2}+n\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.2)</td></tr><tr id="a0000001717"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cot ^2(\theta )+1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \csc ^2(\theta ) \qquad \left(\theta \neq n\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.3)</td></tr></table><p><b class="bfseries">The angle sum formulas</b></p><table id="a0000001718" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001719"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin (a+b)[/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 \sin (a)\cos (b)+\sin (b)\cos (a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.4)</td></tr><tr id="a0000001720"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos (a+b)[/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 \cos (a)\cos (b)-\sin (a)\sin (b)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.5)</td></tr></table><p><b class="bfseries">The double angle formulas</b></p><table id="a0000001721" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001722"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin (2a)[/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 2\sin (a)\cos (a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.6)</td></tr><tr id="a0000001723"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos (2a)[/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 \cos ^2(a)-\sin ^2(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.7)</td></tr></table><p><b class="bfseries">The half angle formulas</b></p><table id="a0000001724" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001725"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sin ^2(a)[/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-\cos (2a)}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.8)</td></tr><tr id="a0000001726"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos ^2(a)[/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+\cos (2a)}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.9)</td></tr></table><p>
We will derive these identities in the exercises below. These exercises are worth zero points, but they will give you a geometric picture for the identities above so that you do not have to blindly memorize them. </p>
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Review: The Pythagorean theorem
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<center><img alt="A right triangle with legs A and B and hypotenuse C is drawn side the arc of a circle whose radius is equal to C. In this case, C equals 1. The angle between the base A and the hypotenuse C is labeled theta." src="/assets/courseware/v1/f9a271ef4748025041d2955f78fe700a/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trig_pyth.svg" style="margin: 10px 25px 25px 25px" width="300px"/><br/>The Pythagorean theorem when hypotenuse is [mathjaxinline]\, 1[/mathjaxinline]: [mathjaxinline]\, \, A^2+B^2=C^2[/mathjaxinline]<br/></center>
<p>
The Pythagorean theorem is equivalent to the following three trigonometric identities. Fill in the blanks. <br/></p>
<p>
<p style="display:inline">[mathjaxinline]\cos ^2(\theta )+\sin ^2(\theta )\, =\,[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_technique1-tab3-problem1_2_1">(*)</span>
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<p>
Divide both sides of the identity (*) above by [mathjaxinline]\, \cos ^2(\theta )\,[/mathjaxinline] to obtain an identity of [mathjaxinline]\, \sec (\theta )\,[/mathjaxinline] and [mathjaxinline]\, \tan (\theta )\,[/mathjaxinline] for [mathjaxinline]\, \theta \neq \frac{\pi }{2}+n\pi[/mathjaxinline]. <br/>(Enter the <b class="bf">left hand side</b> and <b class="bf">right hand side</b> of the resulting identity without switching sides. Your answers should be in terms of [mathjaxinline]\sec (\theta )[/mathjaxinline] and [mathjaxinline]\tan (\theta )[/mathjaxinline] only.)<br/></p>
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[mathjaxinline]=[/mathjaxinline]</td>
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[mathjaxinline]\displaystyle \left(\theta \neq \frac{\pi }{2}+n\pi \right)[/mathjaxinline]</td>
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<p>
Divide both sides of the first identity (*) above (involving [mathjaxinline]\, \cos ^2(\theta )\,[/mathjaxinline] and [mathjaxinline]\, \sin ^2(\theta )\,[/mathjaxinline]) by [mathjaxinline]\, \sin ^2(\theta )\,[/mathjaxinline] to obtain an identity of of [mathjaxinline]\, \csc (\theta )\,[/mathjaxinline] and [mathjaxinline]\, \cot (\theta )\,[/mathjaxinline] for [mathjaxinline]\, \theta \neq n\pi[/mathjaxinline].<br/>(Enter the <b class="bf">left hand side</b> and <b class="bf">right hand side</b> of the resulting identity without switching sides. Your answers should be in terms of [mathjaxinline]\, \csc (\theta )\,[/mathjaxinline] and [mathjaxinline]\, \cot (\theta )\,[/mathjaxinline] only.)<br/></p>
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[mathjaxinline]=[/mathjaxinline]</td>
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[mathjaxinline]\displaystyle \left(\theta \neq n\pi \right)[/mathjaxinline]</td>
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Review: angle sum formulas
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<center><img alt="An arc of circle of radius 1 is shown. The arc is swept by the sum of angles a and b. The green triangle lies in the arc swept by angle a and is a right triangle. Its top corner touches the boundary of the cirlce and has an angle alpha. The vertical leg of this triangle is labeled y. The horizontal leg is labeled x. The hypotenuse is labeled h. The hypotenuse extends until it forms a right angle outside the triangle with the radius formed by the top of angle b. The pink triangle has hypotenuse capital H that extends from the center of the circle and touches the green triangle at the corner between x and h. The pink triangle has vertical leg capital Y and horizontal leg capital X." src="/assets/courseware/v1/cdfb64c47cfdf17d120dbd93a7f4df11/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trig_doubleangle.svg" style="margin: 10px 25px 25px 25px" width="350px"/><br/>Two angles [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b\,[/mathjaxinline] at the center of a circle with radius [mathjaxinline]\, 1[/mathjaxinline] </center>
<p>
Consider the green triangle in the figure above. Find the angle [mathjaxinline]\, \alpha ,\,[/mathjaxinline] and the three edges [mathjaxinline]\, h\,[/mathjaxinline] [mathjaxinline]\, x,\,[/mathjaxinline] [mathjaxinline]\, y\,[/mathjaxinline] of this triangle in terms of the angles [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline].<br/>(Enter your answers in terms of [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline]. Do not use inverse trig functions in your answers.)<br/><p style="display:inline">The angle[mathjaxinline]\, \, \alpha =\, \,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab3-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Here is the same figure as above.<br/></p>
<center><img alt="An arc of circle of radius 1 is shown. The arc is swept by the sum of angles a and b. The green triangle lies in the arc swept by angle a and is a right triangle. Its top corner touches the boundary of the cirlce and has an angle alpha. The vertical leg of this triangle is labeled y. The horizontal leg is labeled x. The hypotenuse is labeled h. The hypotenuse extends until it forms a right angle outside the triangle with the radius formed by the top of angle b. The pink triangle has hypotenuse capital H that extends from the center of the circle and touches the green triangle at the corner between x and h. The pink triangle has vertical leg capital Y and horizontal leg capital X." src="/assets/courseware/v1/cdfb64c47cfdf17d120dbd93a7f4df11/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_trig_doubleangle.svg" style="margin: 10px 25px 25px 25px" width="350px"/><br/>Two angles [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b\,[/mathjaxinline] at the center of a circle with radius [mathjaxinline]\, 1[/mathjaxinline] </center>
<p>
Now, consider the pink triangle. Find the three edges [mathjaxinline]\, H\,[/mathjaxinline] [mathjaxinline]\, X,\,[/mathjaxinline] [mathjaxinline]\, Y\,[/mathjaxinline] of this triangle in terms of the angles [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline].<br/>(Enter your answers in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]H\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="formulaequationinput_technique1-tab3-problem2_6_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Identify the edges with lengths [mathjaxinline]\, \sin (a+b)\,[/mathjaxinline] and [mathjaxinline]\, \cos (a+b)\,[/mathjaxinline] in the figure, and find [mathjaxinline]\, \sin (a+b)\,[/mathjaxinline] and [mathjaxinline]\, \cos (a+b)\,[/mathjaxinline] in terms of the lengths [mathjaxinline]\, \, x[/mathjaxinline],[mathjaxinline]\, \, y[/mathjaxinline],[mathjaxinline]\, \, h[/mathjaxinline], [mathjaxinline]\, \, X[/mathjaxinline], [mathjaxinline]\, \, Y[/mathjaxinline], [mathjaxinline]\, \, H[/mathjaxinline].<br/>(Enter your answers in terms of [mathjaxinline]\, \, x[/mathjaxinline],[mathjaxinline]\, \, y[/mathjaxinline],[mathjaxinline]\, \, h[/mathjaxinline], [mathjaxinline]\, \, X[/mathjaxinline], [mathjaxinline]\, \, Y[/mathjaxinline], [mathjaxinline]\, \, H[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]\sin (a+b)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="formulaequationinput_technique1-tab3-problem2_9_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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</div></div><br/><p style="display:inline">[mathjaxinline]\cos (a+b)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 9" role="group"><div id="formulaequationinput_technique1-tab3-problem2_10_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Finally, express [mathjaxinline]\, \sin (a+b)\,[/mathjaxinline] and [mathjaxinline]\, \cos (a+b)\,[/mathjaxinline] in terms of the angles [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline].<br/>(Enter your answers in terms of [mathjaxinline]\, a\,[/mathjaxinline] and [mathjaxinline]\, b[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]\sin (a+b)\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 10" role="group"><div id="formulaequationinput_technique1-tab3-problem2_11_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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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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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
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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>
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<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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Set [mathjaxinline]a=b[/mathjaxinline] in the angle sum formulas in the previous problem to get the double angle formulas.<br/></p>
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<p style="display:inline">[mathjaxinline]\sin (2a)=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\cos (2a)=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<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>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Review: half angle formulas
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We will now derive the half angle formulas from the double angle formulas.<br/></p>
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Use the identity [mathjaxinline]\, \, \cos ^2(a)+\sin ^2(a)=1[/mathjaxinline] to express [mathjaxinline]\cos (2a)[/mathjaxinline] in terms of [mathjaxinline]\sin (a)[/mathjaxinline] only.<br/><p style="display:inline">[mathjaxinline]\cos (2a)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab3-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Hence, write [mathjaxinline]\sin ^2(a)[/mathjaxinline] in terms of [mathjaxinline]\cos (2a)[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]\sin ^2(a)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique1-tab3-problem4_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Use [mathjaxinline]\cos ^2(a)+\sin ^2(a)=1[/mathjaxinline] to express [mathjaxinline]\cos (2a)[/mathjaxinline] in terms of [mathjaxinline]\cos (a)[/mathjaxinline] only.<br/><p style="display:inline">[mathjaxinline]\cos (2a)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_technique1-tab3-problem4_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Hence,[mathjaxinline]\cos ^2(a)[/mathjaxinline] in terms of [mathjaxinline]\cos (2a)[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]\cos ^2(a)=[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_technique1-tab3-problem4_5_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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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">sin, cos, tan, sec, csc, cot</td>
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Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Review: when the exponent of cosine is one
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Use the method of substitution to evaluate [mathjaxinline]\displaystyle \int \cos (y)\sin ^3(y) \, dy[/mathjaxinline].<br/></p>
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(Use [mathjaxinline]C[/mathjaxinline] as the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \cos (y)\sin ^3(y) \, dy\, =\,[/mathjaxinline]</p>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
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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>
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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/>
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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">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>
<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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<h3 class="hd hd-2">Integrals of sines and cosines: example 1</h3>
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Review: when the exponent of sine is one
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Let [mathjaxinline]m\neq -1[/mathjaxinline]. Evaluate the [mathjaxinline]\displaystyle \int \cos ^ m(x)\sin (x) \, dx[/mathjaxinline].<br/></p>
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(Use [mathjaxinline]C[/mathjaxinline] as the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \cos ^ m(x)\sin (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">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">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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<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">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">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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<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 tangent times secant
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Find the exponents [mathjaxinline]n[/mathjaxinline] and [mathjaxinline]m[/mathjaxinline] such that [mathjaxinline]\, \, \displaystyle \tan (x)\sec (x)\, =\, \cos ^ m(x)\sin ^ n(x)[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]n\, \, =\, \,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab4-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Apply the formula from the previous problem to evaluate [mathjaxinline]\, \, \int \tan (x)\sec (x)\, dx[/mathjaxinline].<br/>(Use [mathjaxinline]C[/mathjaxinline] as the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \tan (x)\sec (x)\, dx\, \, =\, \,[/mathjaxinline]</p>
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Integral of tangent
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<p><p style="display:inline">[mathjaxinline]\displaystyle \int \tan (w)\, dw\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_technique1-tab4-problem4_2_1" class="text-input-dynamath capa_inputtype inline textline">
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Integral of cotangent
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Use FTC1 to evaluate [mathjaxinline]\, \displaystyle \int _{\pi /4}^{\pi /2} \cot (y)\, dy[/mathjaxinline].<br/>(You can enter the functional form of your numerical answer, e.g. enter &#8220;tan(pi/3)" for [mathjaxinline]\tan (\pi /3)[/mathjaxinline].)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{\pi /4}^{\pi /2} \cot (y)\, dy\, =\,[/mathjaxinline]</p>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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Higher odd exponent of sine
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Evaluate [mathjaxinline]\displaystyle \, \, \int \sin ^5(\theta )\cos ^2(\theta ) \, d\theta \,[/mathjaxinline] by first rewriting the integrand in the form [mathjaxinline]\, \sin (\theta ) \, P(\cos (\theta ))\,[/mathjaxinline] where [mathjaxinline]\, P(\cos (\theta ))\,[/mathjaxinline] is a polynomial in [mathjaxinline]\, \cos (\theta )[/mathjaxinline].<br/></p>
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(Use [mathjaxinline]C[/mathjaxinline] as the constant of integration.)<br/></p>
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<p style="display:inline">Hence, [mathjaxinline]\, \, \displaystyle \int \sin ^5(\theta )\cos ^2(\theta ) \, d\theta \, =\,[/mathjaxinline]</p>
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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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Odd power of cosine
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Evaluate [mathjaxinline]\displaystyle \, \, \int \cos ^3(\theta ) \, d\theta[/mathjaxinline] by first rewriting the integrand in the form [mathjaxinline]\, \cos (\theta ) \, P(\sin (\theta ))\,[/mathjaxinline] where [mathjaxinline]\, P(\sin (\theta ))\,[/mathjaxinline] is a polynomial in [mathjaxinline]\, \sin (\theta )[/mathjaxinline].<br/></p>
<p>
(Use [mathjaxinline]C[/mathjaxinline] as the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \cos ^3(\theta ) \, d\theta \, =\,[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
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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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<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 <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>
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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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Odd power of cosine divided by a power of sine
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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 \cot (\theta )\cos ^2(\theta )\, d\theta \, =\,[/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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<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Two odd exponents
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^{\pi /2} \sin ^3(\theta )\cos ^9(\theta )\, d\theta \, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">6. Summarizing integrals with a positive odd exponent</h2>
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Let [mathjaxinline]\, 2m+1\,[/mathjaxinline] be any odd, positive integer and [mathjaxinline]\, n\,[/mathjaxinline] any real number. We can use substitution to evaluate the following: </p><table id="a0000001808" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001809"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \, \cos ^{2m+1} (x) \sin ^ n(x)\, dx[/mathjaxinline]
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[mathjaxinline]\displaystyle :[/mathjaxinline]
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[mathjaxinline]\displaystyle \text {use}\, \, \, \, u=\sin (x)[/mathjaxinline]
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[mathjaxinline]\displaystyle \int \, \sin ^{2m+1} (x) \cos ^ n(x)\, dx[/mathjaxinline]
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[mathjaxinline]\displaystyle :[/mathjaxinline]
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[mathjaxinline]\displaystyle \text {use}\, \, \, \, u=\cos (x).[/mathjaxinline]
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<p><b class="bfseries">The half angle formulas</b></p><p>
Recall </p><table id="a0000001824" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001825"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sin ^2(a)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{1-\cos (2a)}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.81)</td></tr><tr id="a0000001826"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos ^2(a)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac{1+\cos (2a)}{2}.[/mathjaxinline]
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Integral of sine squared
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<p><p style="display:inline">[mathjaxinline]\displaystyle \int \sin ^2(y) \, dy\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab7-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h3 class="hd hd-2">Worked example: trig integrals and a volume of revolution</h3>
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<h2 class="hd hd-2 unit-title">8. Even positive exponent</h2>
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<p><b class="bfseries">Integral with two even positive exponents</b></p><p>
To evaluate </p><table id="a0000001830" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001831"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sin ^{2m}(x) \cos ^{2n}(x) \, dx \, \, \qquad 2m,2n\geq 0 \, \, \text {even integers},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.85)</td></tr></table><p>
we apply the half angle formulas to the integrand as many times as needed, until the integrand does not contain any non-constant term in which both exponents of [mathjaxinline]\, \sin \,[/mathjaxinline] and [mathjaxinline]\, \cos \,[/mathjaxinline] are even, but instead may contain terms with odd powers of [mathjaxinline]\, \sin (2kx)\,[/mathjaxinline] and [mathjaxinline]\, \cos (2lx)\,[/mathjaxinline] where [mathjaxinline]\, 2k\,[/mathjaxinline] and [mathjaxinline]\, 2l\,[/mathjaxinline] are even integers.<br/></p><p>
We will find a special kind of general formulas for these integrals when we discuss integration by parts later.<br/></p>
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Using sine of double angles
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Find [mathjaxinline]\, n\,[/mathjaxinline] and [mathjaxinline]\, a\,[/mathjaxinline] such that [mathjaxinline]\displaystyle \sin ^4(3x) \cos ^4(3x) \, =\, \frac{\sin ^ n(ax)}{2^4}[/mathjaxinline]. <br/><p style="display:inline">[mathjaxinline]n\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab8-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Even power of cosine
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This is one of the most computationally intensive integrals we will do in this section.<br/></p>
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We will derive formulas for integrals of even power of cosine later in this unit.<br/></p>
<p><p style="display:inline">[mathjaxinline]\displaystyle \int _0^{\pi /4} \cos ^6(x) \, dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab8-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Review: derivative of cotangent
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dy}\cot (y)\, =\,[/mathjaxinline]</p>
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Review: derivative of cosecant
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{d}{dy}\csc (y)\, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">10. Integrals with an odd positive power of tangent</h2>
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Integrals of the form </p><table id="a0000001855" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001856"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \tan ^{2m+1}(x)\, \sec ^ n(x)\, dx \qquad \text {with}\, \, n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd integer}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.103)</td></tr><tr id="a0000001857"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot ^{2m+1}(x)\, \csc ^ n(x)\, dx \qquad \text {with}\, \, n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd integer}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.104)</td></tr></table><p>
are not new, since</p><table id="a0000001858" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001859"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \tan ^{2m+1}(x)\, \sec ^{n}(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 \sin ^{2m+1}(x)\, \cos ^{-(n+2m+1)}(x)\qquad (2m+1\geq 1 \, \, \text {odd integer})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.105)</td></tr><tr id="a0000001860"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cot ^{2m+1}(x) \, \csc ^ n(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \cos ^{2m+1}(x)\, \sin ^{-(n+2m+1)}(x)\qquad (2m+1\geq 1 \, \, \text {odd integer}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.106)</td></tr></table><p>
and we already know how to evaluate these integrals using the substitutions [mathjaxinline]\, u=\cos (x)\,[/mathjaxinline] and [mathjaxinline]\, u=\sin (x)\,[/mathjaxinline] respectively.<br/></p><p>
We now use alternate substitutions:<br/></p><table id="a0000001861" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001862"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \tan ^{2m+1}(x) \sec ^ n(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 \text {use}\, \, u=\sec (x)\qquad (2m+1\geq 1 \, \, \text {odd integer})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.107)</td></tr><tr id="a0000001863"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot ^{2m+1}(x) \csc ^ n(x)\, dx[/mathjaxinline]
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[mathjaxinline]\displaystyle :[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {use}\, \, u=\csc (x) \qquad (2m+1\geq 1 \, \, \text {odd integer}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.108)</td></tr></table>
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Use [mathjaxinline]\displaystyle \, 1+\tan ^2(x)=\sec ^2(x)\,[/mathjaxinline] and the substitution [mathjaxinline]u=\sec (x)\,[/mathjaxinline] to evaluate [mathjaxinline]\, \displaystyle \int \tan ^3(x)\sec (x)\, dx[/mathjaxinline].<br/>(Use [mathjaxinline]\, C[/mathjaxinline] as the constant of integration.)<br/><p style="display:inline">[mathjaxinline]\displaystyle \int \tan ^3(x)\sec (x)\, dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab10-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">11. Integral of secant and cosecant</h2>
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<p><b class="bfseries"><span style="color:#FF7F00">Note on video:</span></b> Note that there is a missing absolute value sign in the video below. We will repeat the result of the video immediately afterwards.<br/></p>
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<p><b class="bfseries">Integral of secant</b></p><p>
Observe that </p><table id="a0000001896" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001897"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{d}{dx} \left(\sec (x)+\tan (x)\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 \sec (x)\tan (x)+\sec ^2(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.132)</td></tr><tr id="a0000001898"><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 \sec (x)\, \left(\tan (x)+\sec (x)\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.133)</td></tr></table><p>
This implies </p><table id="a0000001899" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001900"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sec (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 \frac{u'}{u}\qquad \text {where }\, \, u=\sec (x)+\tan (x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.134)</td></tr></table><p>
Integrating both sides, we get </p><table id="a0000001901" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001902"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sec (x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \int \frac{du}{u}\qquad \text {where }\, \, u=\sec (x)+\tan (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.135)</td></tr><tr id="a0000001903"><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 \ln |u|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.136)</td></tr><tr id="a0000001904"><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 \ln |\sec (x)+\tan (x)|+C.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.137)</td></tr></table><p>
(The absolute value signs in the last two lines were missing in the video.)<br/></p>
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Integral of cosecant
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Let [mathjaxinline]\, 2m\geq 2\,[/mathjaxinline] be a positive even integer, and [mathjaxinline]\, n\,[/mathjaxinline] any real number.<br/></p><p>
To evaluate integrals with an even positive power of [mathjaxinline]\sec[/mathjaxinline], </p><table id="a0000001910" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001911"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sec ^{2m}(x)\tan ^ n(x)\, dx\qquad \left(2m\geq 2\, \, \text {positive integer},\, \, n \, \text {real}\right),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.141)</td></tr></table><p>
we can substitution with [mathjaxinline]\, u=\tan (x),\,[/mathjaxinline] with [mathjaxinline]\, du=\sec ^2(x)\, dx,\,[/mathjaxinline]and then use the identity </p><table id="a0000001912" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001913"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sec ^2(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 1+\tan ^2(x)\qquad \left(x\neq \frac{\pi }{2}+k\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.142)</td></tr></table><p>
to rewrite the integrand in terms of only [mathjaxinline]\, u[/mathjaxinline].<br/></p><p>
Similarly, to evaluate integral with even positive power of [mathjaxinline]\, \csc[/mathjaxinline], </p><table id="a0000001914" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001915"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \csc ^{2m}(x)\cot ^ n(x)\, dx\qquad \left(2m\geq 2\, \, \text {positive integer},\, \, n \, \text {real}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.143)</td></tr><tr id="a0000001916"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.144)</td></tr></table><p>
we can use the substitution [mathjaxinline]\, u=\cot (x)\,[/mathjaxinline] and the identity </p><table id="a0000001917" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001918"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \csc ^2(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \cot ^2(x)+1\qquad \left(x\neq n\pi \right).[/mathjaxinline]
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Practice 2
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<p><p style="display:inline">[mathjaxinline]\displaystyle \int \sec ^2(x) \tan ^{4}(x)\, dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab12-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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</div></div><br/>(Use [mathjaxinline]\, C\,[/mathjaxinline] as the constant of integration.)<br/></p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
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<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>
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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>
<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>
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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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<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>
</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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Practice 3
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<p><p style="display:inline">[mathjaxinline]\displaystyle \int \cot ^4(y)\, dy\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab12-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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\(\)
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</div></div><br/>(Use [mathjaxinline]\, C\,[/mathjaxinline] as the constant of integration.)<br/>(Simplify your answer so that it does not contain inverse trig functions ([mathjaxinline]\arctan[/mathjaxinline] etc).)<br/></p>
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<tr class="fiptitle">
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<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>
<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>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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So far we know how to integrate </p><table id="a0000001931" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001932"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sin ^ n(x) \cos ^ m(x)\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.155)</td></tr></table><p>
when </p><ol class="enumerate"><li value="1"><p>
one of [mathjaxinline]\, n\,[/mathjaxinline] and [mathjaxinline]\, m\,[/mathjaxinline] is an odd positive integer (and the other can be any real number); </p></li><li value="2"><p>
both [mathjaxinline]\, n\,[/mathjaxinline] and [mathjaxinline]\, m\,[/mathjaxinline] are even positive integers. </p></li></ol><p>
The integrals we just discussed, </p><table id="a0000001933" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001934"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sec ^{2m}(x)\tan ^ n(x)\, dx\qquad \left(2m\geq 2\, \, \text {positive integer},\, \, n \, \text {real}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.156)</td></tr><tr id="a0000001935"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \csc ^{2m}(x)\cot ^ n(x)\, dx\qquad \left(2m\geq 2\, \, \text {positive integer},\, \, n \, \text {real}\right),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.157)</td></tr></table><p>
can be rewritten into [mathjaxinline]\, \displaystyle \int \sin ^ n(x) \cos ^ m(x)\, dx.\, \,[/mathjaxinline] What are the conditions on the exponents in this new case? What new integrals can we now evaluate? We will think about this in the problems below.<br/></p>
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Integrand in terms of powers of cosine and sine
</h3>
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<p>
Let [mathjaxinline]\, 2m\geq 2\,[/mathjaxinline] be a (positive) even integer and [mathjaxinline]\, n\,[/mathjaxinline] any real number.<br/></p>
<p>
Which of the following can be rewritten as [mathjaxinline]\, \sec ^{2m}(x)\tan ^ n(x)[/mathjaxinline]? <br/>(Consider both when [mathjaxinline]\, n\, \geq 0[/mathjaxinline] and [mathjaxinline]\, n&lt;0.[/mathjaxinline])<br/>(Check all that apply.)<br/></p>
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<text>[mathjaxinline]\displaystyle \frac{\sin ^7(x)}{\cos ^3(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{\sin ^3(x)}{\cos ^7(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{\sqrt {\sin (x)}}{\sqrt {\cos ^{9}(x)}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{1}{\sin ^2(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{1}{\sqrt {\sin (x)}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{\cos (x)}{\sin ^3(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{\cos ^{1/3}(x)}{\sin ^{25/3}(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{\cos ^{1/2}(x)}{\sin ^{3/2}(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{1}{\sin ^2(x)\cos ^2(x)}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{1}{\sin (x)\cos ^2(x)}[/mathjaxinline]</text>
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Secant squared In terms of cosecant and cotangent
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Find the exponents [mathjaxinline]\, p\,[/mathjaxinline] and [mathjaxinline]\, q\,[/mathjaxinline] such that [mathjaxinline]\, \displaystyle \sec ^2(x)\, =\, \cot ^ p(x)\csc ^ q(x)[/mathjaxinline].<br/><p style="display:inline">[mathjaxinline]p\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab13-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Products of powers of cosecant and cotangent
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Find [mathjaxinline]\, p\,[/mathjaxinline] and [mathjaxinline]\, q\,[/mathjaxinline] in terms of [mathjaxinline]\, m\,[/mathjaxinline] and [mathjaxinline]\, n\,[/mathjaxinline] such that [mathjaxinline]\displaystyle \tan ^ n(x)\sec ^{2m}(x)\, =\, \cot ^ p(x)\csc ^ q(x).[/mathjaxinline]<br/>(Enter your answer in terms of [mathjaxinline]\, m\,[/mathjaxinline] and [mathjaxinline]\, n[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]p\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab13-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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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">
<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>
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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>
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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>
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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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<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>
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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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Evaluate [mathjaxinline]\, \, \displaystyle \int \frac{1}{\sin ^4(x)\cos ^2(x)}\, dx[/mathjaxinline].<br/>(Use [mathjaxinline]\, C\,[/mathjaxinline] as the constant of integration.)<br/><p style="display:inline">[mathjaxinline]\displaystyle \int \frac{1}{\sin ^4(x)\cos ^2(x)}\, dx\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_technique1-tab14-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<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>
<td class="formulainput">
<font color="#0078b0">2520</font>
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</tr>
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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>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Practice 2
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<p style="display:inline">[mathjaxinline]\displaystyle \int ^{\pi /3}_{\pi /4} \frac{1}{\sin (x)\cos (x)}\, dx\, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">15. Higher power in numerator</h2>
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So far, we can evaluate integrals involving fractions of powers of [mathjaxinline]\, \cos \,[/mathjaxinline] and [mathjaxinline]\, \sin ,\,[/mathjaxinline]<br/></p><table id="a0000001980" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001981"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \frac{\sin ^ n(x)}{\cos ^ k(x)}\, dx\qquad \text {with}\, \, n,k \geq 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.187)</td></tr><tr id="a0000001982"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \frac{\cos ^ n(x)}{\sin ^ k(x)}\, dx \qquad \text {with}\, \, n,k \geq 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.188)</td></tr></table><p>
in the following two cases: </p><ol class="enumerate"><li value="1"><p>
When [mathjaxinline]\, n\,[/mathjaxinline] is a positive odd integer, use the substitutions [mathjaxinline]\, u=\cos (x)\,[/mathjaxinline] for [mathjaxinline]\displaystyle \, \int \frac{\sin ^ n(x)}{\cos ^ k(x)},\,[/mathjaxinline] and [mathjaxinline]\, u=\sin (x)\,[/mathjaxinline] for [mathjaxinline]\displaystyle \, \int \frac{\cos ^ n(x)}{\sin ^ k(x)}[/mathjaxinline].<br/></p></li><li value="2"><p>
When [mathjaxinline]\, n-k\,[/mathjaxinline] is a negative even integer (so in particular [mathjaxinline]\, k>n\,[/mathjaxinline]), rewrite the integrals as [mathjaxinline]\, \displaystyle \displaystyle \int \tan ^ n(x)\sec ^{2m}(x)\, dx\,[/mathjaxinline] (or as [mathjaxinline]\, \displaystyle \int \cot ^ n(x)\csc ^{2m}(x)\, dx\,[/mathjaxinline]), and use the substitution [mathjaxinline]\, u=\tan (x)\,[/mathjaxinline] (or [mathjaxinline]\, u=\cot (x)[/mathjaxinline]). </p></li></ol><p>
These do not cover the case when the exponent in the numerator is greater than the exponent in the denominator, </p><table id="a0000001983" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001984"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \frac{\sin ^ n(x)}{\cos ^ k(x)}\, dx\qquad \text {with}\, \, n>k > 0,[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.189)</td></tr><tr id="a0000001985"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \frac{\cos ^ n(x)}{\sin ^ k(x)}\, dx \qquad \text {with}\, \, n>k > 0.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.190)</td></tr></table><p>
Let us end this section with one of these integrals.<br/></p>
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Higher exponent in the numerator
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Evaluate [mathjaxinline]\, \displaystyle \int \tan ^2(y) \sin ^2(y)\, dy\,[/mathjaxinline] by first rewriting the integrand as a polynomial in [mathjaxinline]\, \cos (y)\,[/mathjaxinline] plus a polynomial in [mathjaxinline]\, \sec (y)[/mathjaxinline].<br/>(Use [mathjaxinline]\, C\,[/mathjaxinline] as the constant of integration.)<br/></p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \tan ^2(y) \sin ^2(y)\, dy\, =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">16. Summary</h2>
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<p><b class="bfseries">Trigonometric identities</b></p><p><b class="bfseries">The Pythagorean theorem</b></p><table id="a0000001991" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001992"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos ^2(\theta )+\sin ^2(\theta )[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.195)</td></tr><tr id="a0000001993"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle 1+\tan ^2(\theta )[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sec ^2(\theta )\qquad \left(\theta \neq \frac{\pi }{2}+n\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.196)</td></tr><tr id="a0000001994"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cot ^2(\theta )+1[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \csc ^2(\theta ) \qquad \left(\theta \neq n\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.197)</td></tr></table><p><b class="bfseries">The angle sum formulas</b></p><table id="a0000001995" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001996"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin (a+b)[/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 \sin (a)\cos (b)+\sin (b)\cos (a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.198)</td></tr><tr id="a0000001997"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos (a+b)[/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 \cos (a)\cos (b)-\sin (a)\sin (b)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.199)</td></tr></table><p><b class="bfseries">The double angle formulas</b></p><table id="a0000001998" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001999"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin (2a)[/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 2\sin (a)\cos (a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.200)</td></tr><tr id="a0000002000"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos (2a)[/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 \cos ^2(a)-\sin ^2(a)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.201)</td></tr></table><p><b class="bfseries">The half angle formulas</b></p><table id="a0000002001" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002002"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sin ^2(a)[/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-\cos (2a)}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.202)</td></tr><tr id="a0000002003"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos ^2(a)[/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+\cos (2a)}{2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.203)</td></tr></table><p><b class="bfseries">Integrals of products of powers of sin and cosine</b></p><p><b class="bfseries">One exponent equals one</b></p><table id="a0000002004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \cos ^ m(x)\sin (x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\begin{cases} \displaystyle -\frac{\cos (x)^{m+1}}{m+1}+C, & \mbox{if } m\neq -1 \\ -\ln \left|\cos (x)\right|+C & \mbox{if } m=-1. \end{cases}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.204)</td></tr><tr id="a0000002006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sin ^ m(x)\cos (x) \, dx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\begin{cases} \displaystyle \frac{\sin (x)^{m+1}}{m+1}+C, & \mbox{if } m\neq -1 \\ \ln \left|\sin (x)\right|+C & \mbox{if } m=-1. \end{cases}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.205)</td></tr></table><p>
The formulas for [mathjaxinline]m=-1,\, \,[/mathjaxinline] are equivalent to </p><table id="a0000002007" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \tan (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 -\ln \left|\cos (x)\right|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.206)</td></tr><tr id="a0000002009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot (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 \ln \left|\sin (x)\right|+C.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.207)</td></tr></table><p>
The formulas for [mathjaxinline]m=-2,\, \,[/mathjaxinline] are equivalent to </p><table id="a0000002010" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002011"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \tan (x)\sec (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 \sec (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.208)</td></tr><tr id="a0000002012"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot (x)\csc (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 -\csc (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.209)</td></tr></table><p>
Note the sign changes in all the formulas above when we switch [mathjaxinline]\sin[/mathjaxinline] and [mathjaxinline]\cos[/mathjaxinline].<br/></p><p><b class="bfseries">One positive odd exponent</b></p><p>
Let [mathjaxinline]\, 2m+1\,[/mathjaxinline] be any positive integer and [mathjaxinline]\, n\,[/mathjaxinline] any real number. We can use substitution to evaluate the following: </p><table id="a0000002013" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002014"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \, \cos ^{2m+1} (x) \sin ^ n(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 \text {use}\, \, \, \, u=\sin (x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.210)</td></tr><tr id="a0000002015"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \, \sin ^{2m+1} (x) \cos ^ n(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 \text {use}\, \, \, \, u=\cos (x).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.211)</td></tr></table><p><b class="bfseries">Two positive even exponents</b></p><p>
To evaluate </p><table id="a0000002016" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002017"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sin ^{2m}(x) \cos ^{2n}(x) \, dx \, \, \qquad 2m,2n\geq 0 \, \, \text {even integers},[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.212)</td></tr></table><p>
we apply the half angle formulas to the integrand as many times as needed, until the integrand does not contain any non-constant term in which both exponents of [mathjaxinline]\, \sin \,[/mathjaxinline] and [mathjaxinline]\, \cos \,[/mathjaxinline] are even, but instead may contain terms with odd powers of [mathjaxinline]\, \sin (2kx)\,[/mathjaxinline] and [mathjaxinline]\, \cos (2lx)\,[/mathjaxinline] where [mathjaxinline]\, 2k\,[/mathjaxinline] and [mathjaxinline]\, 2l\,[/mathjaxinline] are even integers.<br/></p><p>
We will find a special kind of general formulas for these integrals when we discuss integration by parts later.<br/></p><p><b class="bfseries">Integrals of products of powers of tangent and cosecant</b></p><p><b class="bfseries">Odd positive exponent of tangent</b></p><p>
The integrals of the form </p><table id="a0000002018" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002019"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \tan ^{2m+1}(x) \sec ^ n(x)\, dx \qquad (n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd positive integer})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.213)</td></tr><tr id="a0000002020"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot ^{2m+1}(x) \csc ^ n(x)\, dx \qquad (n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd positive integer})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.214)</td></tr></table><p>
can be evaluated using the substitution [mathjaxinline]\, u=\sec (x),\, \,[/mathjaxinline] and [mathjaxinline]\, u=\csc (x)\,[/mathjaxinline] respectively.<br/></p><p>
Alternatively, since these integrals can also be written as </p><table id="a0000002021" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002022"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sin ^{2m+1}(x)\cos ^{-n}(x)\, dx \qquad (n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd positive integer})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.215)</td></tr><tr id="a0000002023"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \cos ^{2m+1}(x)\sin ^{-n}(x)\, dx\qquad (n\, \text {real},\, \, 2m+1\geq 1 \, \, \text {odd positive integer}),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.216)</td></tr></table><p>
we can also use evaluate them using the substitutions [mathjaxinline]\, u=\cos (x)\,[/mathjaxinline] and [mathjaxinline]\, u=\sin (x)\,[/mathjaxinline] respectively.<br/></p><p><b class="bfseries">Even positive exponent of secant</b></p><p>
Let [mathjaxinline]\, 2m\geq 2\,[/mathjaxinline] be a positive even integer, and [mathjaxinline]\, n\,[/mathjaxinline] any real number.<br/></p><p>
We can use [mathjaxinline]\, u=\, \tan (x)\,[/mathjaxinline] to evaluate </p><table id="a0000002024" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002025"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sec ^{2m}(x)\tan ^ n(x)\, dx \qquad (2m\geq 2\, \text {even integer},\, n\, \text { real}),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.217)</td></tr></table><p>
and use [mathjaxinline]\, u=\cot (x)\,[/mathjaxinline] to evaluate </p><table id="a0000002026" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002027"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \csc ^{2m}(x)\cot ^ n(x)\, dx \qquad (2m\geq 2\, \text {even integer},\, n\, \text { real}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.218)</td></tr></table><p>
These integrals are equivalent to </p><table id="a0000002028" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002029"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int \sin ^{p}(x) \cos ^{q}(x) \, dx \, \, \qquad \text {for}\, \, p,q\, \, \text {real}, \, \, \, p+q=-2k \, \, \text {even negative integer}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.219)</td></tr></table><p><b class="bfseries">Some integrals evaluated in this section</b></p><table id="a0000002030" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002031"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sec ^2(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 \tan (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.220)</td></tr><tr id="a0000002032"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \csc ^2(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 -\cot (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.221)</td></tr><tr id="a0000002033"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sec (x)\tan (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 \sec (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.222)</td></tr><tr id="a0000002034"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \csc (x)\cot (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 -\csc (x)+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.223)</td></tr><tr id="a0000002035"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \tan (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 -\ln |\cos (x)|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.224)</td></tr><tr id="a0000002036"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \cot (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 \ln |\sin (x)|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.225)</td></tr><tr id="a0000002037"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sec (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 \ln |\sec (x)+\tan (x)|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.226)</td></tr><tr id="a0000002038"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \csc (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 -\ln |\csc (x)+\cot (x)|+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.227)</td></tr></table>
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