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<h2 class="hd hd-2 unit-title">1. (Optional) Trig integrals and binomial expansion</h2>
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This section is <b class="bf">optional</b>, and is <b class="bf">not graded</b>. You are not held accountable for the content. However, there has already been one comment on the discussion forum about exactly this topic, so we wanted to include it for those who may be interested. </p><p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Find coefficients of any term in a general formula for [mathjaxinline]\, \displaystyle \int \, \cos ^{2m+1} (x) \sin ^ n(x)\, dx,\,[/mathjaxinline] [mathjaxinline]\int \, \sin ^{2m+1} (x) \cos ^ n(x)\, dx,\,[/mathjaxinline] and [mathjaxinline]\int \sec ^{2m}(x) \tan ^ n(x)\, dx[/mathjaxinline]. </p></li></ul><p><b class="bfseries">Contents: 6 pages</b></p><p>
0 videos ( 0 minutes 1x speed) 9 questions </p>
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<h2 class="hd hd-2 unit-title">2. General formula</h2>
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Our goal in the problems below is to find a general formula for [mathjaxinline]\, \, \displaystyle \int \cos ^{2m+1} (x) \sin ^ n(x)\, dx,\, \,[/mathjaxinline] when [mathjaxinline]\, 2m+1\,[/mathjaxinline] is an odd positive integer, and [mathjaxinline],n\neq -(2k+1)[/mathjaxinline] for any integer [mathjaxinline]\, 0\leq k\leq m.\, \,[/mathjaxinline] <br/></p><p>
You will not be held accountable for deriving general formulas, but the following problems should help you read off information you need from a general formula that you look up.<br/></p><p>
We will derive the formula in terms of <span style="color:#27408C"><b class="bf">binomial coefficients</b></span>, which are commonly used. First, we need to introduce factorials.<br/></p><p><b class="bfseries">Factorial</b></p><p>
Let [mathjaxinline]\, m\,[/mathjaxinline] be a non-negative integer. </p><p>
Define the <span style="color:#27408C"><b class="bf">factorial</b></span> of [mathjaxinline]\, m,\,[/mathjaxinline] denoted by [mathjaxinline]\, m!\,[/mathjaxinline] to be </p><table id="a0000002039" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002040"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle m![/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \begin{cases} 1 & \mbox{if } m\, =\, 0\\ m\, (m-1)\, (m-2)\, \cdots \, (3)\, (2)\, (1) & \mbox{if } m\, >0. \end{cases}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.228)</td></tr></table><p>
For example: [mathjaxinline]\, 4!\, =\, (4)\, (3)\, (2)\, (1)\, =\, 24[/mathjaxinline].<br/></p>
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Factorial practice
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Find [mathjaxinline]\, 1!,\,[/mathjaxinline] [mathjaxinline]\, 2!,\,[/mathjaxinline] [mathjaxinline]3![/mathjaxinline].<br/></p>
<p>
<p style="display:inline">[mathjaxinline]1!=\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]2!=\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]3!=\,[/mathjaxinline]</p>
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<p><b class="bfseries">Binomial coefficient</b></p><p>
Let [mathjaxinline]\, m\,[/mathjaxinline] and [mathjaxinline]\, k\,[/mathjaxinline] be non-negative integers, and let [mathjaxinline]\, k\leq m[/mathjaxinline].<br/></p><p>
Define the <span style="color:#27408C"><b class="bf">binomial coefficient "[mathjaxinline]m[/mathjaxinline] choose [mathjaxinline]k,[/mathjaxinline]'"</b></span> denoted by [mathjaxinline]\, \, \displaystyle {m \choose k},\, \,[/mathjaxinline] to be: </p><table id="a0000002045" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002046"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle {m \choose k}[/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{m!}{(m-k)!\, k!}\qquad (\text {"}m\, \text {choose}\, k\text {"})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.232)</td></tr></table><p>
For example, </p><table id="a0000002047" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002048"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle {4 \choose 0}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{4!}{4!\, 0!}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \qquad (0!\, =1)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.233)</td></tr><tr id="a0000002049"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {4 \choose 1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{4!}{3!\, 1!}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{(4)\, (3)\, (2)\, }{(3)\, (2)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 4[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.234)</td></tr><tr id="a0000002050"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {4 \choose 2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{4!}{2!\, 2!}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{(4)\, (3)\, (2)\, }{(2) (2)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 6[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.235)</td></tr><tr id="a0000002051"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {4 \choose 3}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{4!}{1!\, 3!}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{(4)\, (3)\, (2)\, }{(3)\, (2)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 4[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.236)</td></tr><tr id="a0000002052"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle {4 \choose 4}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{4!}{0!\, 4!}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle 1[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.237)</td></tr></table><p>
When [mathjaxinline]\, m\,[/mathjaxinline] is large, we simplify [mathjaxinline]\, \displaystyle {m \choose k} \,[/mathjaxinline] by expanding the factorials and cancelling the same factors in the numerator and denominator.<br/></p><p>
For example, </p><table id="a0000002053" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002054"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle {10 \choose 4}[/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{10!}{6!\, 4!}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.238)</td></tr><tr id="a0000002055"><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 \frac{ (10)\, (9)\, (8)\, (7) \, 6! }{6!\, 4!}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.239)</td></tr><tr id="a0000002056"><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 \frac{ (10)\, (9)\, (8)\, (7)}{(4)\, (3)\, (2)}\qquad (\text {cancel}\, 6!).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.240)</td></tr></table>
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Find the number [mathjaxinline]\displaystyle \, {5 \choose 2}[/mathjaxinline]. </p>
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Find [mathjaxinline]\displaystyle \, {100 \choose 2}[/mathjaxinline].<br/>(You can enter your answer as a fraction of products of numbers, e.g. &#8220;7*6/(3*2)." You answer should not contain factorials.)<br/><p style="display:inline">[mathjaxinline]\, \displaystyle {100 \choose 2} =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab2-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">3. Binomial theorem and the general formula for one odd power</h2>
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<p><b class="bfseries">Binomial theorem</b></p><p>
The <span style="color:#27408C"><b class="bf">binomial theorem</b></span> says that </p><table id="a0000002067" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002068"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle (1+y)^ m[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \sum _{k=0}^{m} {m \choose k} \, y^ k[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \sum _{k=0}^{m} \frac{m!}{(m-k)!\, k!}\, y^ k[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.249)</td></tr></table><p>
In other words, in the expansion of [mathjaxinline]\, (1+y)^ m,\,[/mathjaxinline] the coefficient of [mathjaxinline]\, y^ k\,[/mathjaxinline] is [mathjaxinline]\displaystyle \, {m \choose k}[/mathjaxinline].<br/>For example: the coefficient of [mathjaxinline]\, y^2\,[/mathjaxinline] in [mathjaxinline]\, (1+y)^3\,[/mathjaxinline] is [mathjaxinline]\displaystyle {3\choose 2}\, =\, \frac{3!}{1!\, 2!\, }\, =\, 3,\, \,[/mathjaxinline] as expected.<br/></p>
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Find the coefficient of [mathjaxinline]\, y^3\,[/mathjaxinline] in [mathjaxinline]\, (1+y)^5\,[/mathjaxinline] using the binomial theorem.<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab3-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Find the coefficient of [mathjaxinline]\, y^3\,[/mathjaxinline] in [mathjaxinline]\, (1-y)^{20} \,[/mathjaxinline] using the binomial theorem. <br/><i class="itshape">Hint</i>: Use [mathjaxinline]\, (1-y)\, =\, \left(1+(-y)\right)[/mathjaxinline]. <br/>(You can enter you answer in terms of factorials. Type &#8220;factorial(20)" for [mathjaxinline]\, 20![/mathjaxinline].)<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab3-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">4. General formula for integral of odd powers</h2>
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We now return to find a general formula for [mathjaxinline]\, \, \displaystyle \int \cos ^{2m+1} (x) \sin ^ n(x)\, dx,\, \,[/mathjaxinline] when [mathjaxinline]\, 2m+1\,[/mathjaxinline] is an odd positive integer, and [mathjaxinline]\, n\neq -(2k+1)[/mathjaxinline] for any integer [mathjaxinline]\, 0\leq k\leq m.\, \,[/mathjaxinline] </p>
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Expansion using binomial coefficients
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Let [mathjaxinline]\, m\,[/mathjaxinline] be a positive integer.<br/>Find the coefficient of [mathjaxinline]\, \sin ^{2k}(x)\,[/mathjaxinline] in [mathjaxinline]\displaystyle \left(1-\sin ^2(x)\right)^ m[/mathjaxinline]. Write your answer in terms of factorials.<br/>(Enter <b class="bf">factorial(m)</b> for [mathjaxinline]m![/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]\displaystyle \left(1-\sin ^2(x)\right)^ m\, =\, \sum _{k=0}^ m C_{k} \, \sin ^{2k}(x)\,[/mathjaxinline] for [mathjaxinline]\, C_{k}\, =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab4-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Rewriting the integrand
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Let [mathjaxinline]\, 2m+1\,[/mathjaxinline] be any positive integer and let [mathjaxinline]\, n\,[/mathjaxinline] be any real number.</p>
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To evaluate [mathjaxinline]\, \displaystyle \int \cos ^{2m+1} (x) \sin ^ n(x)\, dx\, \,[/mathjaxinline] using the substitution [mathjaxinline]\, u=\sin (x),\,[/mathjaxinline] we need to first rewrite the integrand as </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000002088" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle \cos ^{2m+1} (x) \sin ^ n(x)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \cos (x) \, f(\sin (x))[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.262)</td>
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where [mathjaxinline]\, f(\sin (x))\,[/mathjaxinline] is a function of [mathjaxinline]\, \sin (x)[/mathjaxinline].<br/></p>
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Find the lowest, second lowest, and highest exponents of [mathjaxinline]\, \sin (x)\,[/mathjaxinline] in [mathjaxinline]\, f(\sin (x))[/mathjaxinline].<br/>(Enter the exponents only, in terms of [mathjaxinline]\, m\,[/mathjaxinline] and [mathjaxinline]\, n[/mathjaxinline].)<br/><p style="display:inline">The <b class="bfseries">lowest</b> exponent of [mathjaxinline]\, \sin (x)\,[/mathjaxinline] in [mathjaxinline]\, f(\sin (x))\,[/mathjaxinline] is </p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab4-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p style="display:inline">The <b class="bfseries">second lowest</b> exponent of [mathjaxinline]\, \sin (x)\,[/mathjaxinline] in [mathjaxinline]\, f(\sin (x))\,[/mathjaxinline] is </p>
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<p style="display:inline">The <b class="bfseries">highest</b> exponent of [mathjaxinline]\, \sin (x)\,[/mathjaxinline] in [mathjaxinline]\, f(\sin (x))\,[/mathjaxinline] is </p>
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Coefficients in the general formula
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As above, let [mathjaxinline]\, 2m+1\,[/mathjaxinline] be an odd positive integer. Let [mathjaxinline]n[/mathjaxinline] be any real number such that [mathjaxinline]\, n \neq -(2k+1)\,[/mathjaxinline] for all integers [mathjaxinline]\, 0\leq k\leq m[/mathjaxinline]. <br/></p>
<p>
Evaluate [mathjaxinline]\, \displaystyle \int \cos ^{2m+1} (x) \sin ^ n(x)\, dx\,[/mathjaxinline] by first rewriting the integrand as [mathjaxinline]\cos \, f(\sin (x))\,[/mathjaxinline] as in the previous problem and then using the substitution [mathjaxinline]\, u=\sin (x)[/mathjaxinline].<br/></p>
<p>
For [mathjaxinline]\, m=100,\,[/mathjaxinline] what is the coefficient of [mathjaxinline]\, \sin ^{n+7}(x)\,[/mathjaxinline] In the formula you obtained for [mathjaxinline]\, \displaystyle \int \cos ^{2m+1} (x) \sin ^ n(x)\, dx\,[/mathjaxinline] (after integrating)?<br/>(Enter your answer in terms of [mathjaxinline]n[/mathjaxinline]).<br/><p style="display:inline">The coefficient of [mathjaxinline]\, \sin ^{n+7}(x)\,[/mathjaxinline] in the evaluation of [mathjaxinline]\, \displaystyle \int \cos ^{201} (x) \sin ^ n(x)\, dx\,[/mathjaxinline] is </p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_extra-tab4-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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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/>
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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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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">5. Integral of even powers of secant</h2>
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General formula for integral of even powers of secant
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Consider the formula for [mathjaxinline]\, \displaystyle \int \sec ^{42}(x)\, dx\,[/mathjaxinline] in terms of only powers of [mathjaxinline]\, \tan ,\,[/mathjaxinline] obtained by using the substitution [mathjaxinline]\, u=\tan (x)[/mathjaxinline].<br/></p>
<p>
Find the coefficients of [mathjaxinline]\, \tan ^6(x)\,[/mathjaxinline] and [mathjaxinline]\, \tan ^7(x)\,[/mathjaxinline] in this formula (after integrating).<br/></p>
<p>
(Enter the coefficients only.)<br/></p>
<p>
<p style="display:inline">The coefficient of [mathjaxinline]\, \tan ^6(x)\,[/mathjaxinline] is </p>
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<p style="display:inline">The coefficient of [mathjaxinline]\, \tan ^7(x)\,[/mathjaxinline] is</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>
<td class="formulainput">Integers</td>
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<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</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>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">6. Summary</h2>
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<p><b class="bfseries">Binomial coefficients</b></p><p>
The <span style="color:#27408C"><b class="bf">binomial theorem</b></span> says that </p><table id="a0000002108" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002109"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle (1+y)^ m[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \sum _{k=0}^{m} {m \choose k} \, y^ k[/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 \sum _{k=0}^{m} \frac{m!}{(m-k)!\, k!}\, y^ k[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.276)</td></tr></table><p>
where [mathjaxinline]\, \, m!\, =\, m\, (m-1)\, (m-2)\, \cdots \, (3)\, (2)\, (1)\,[/mathjaxinline] is called [mathjaxinline]\, m\,[/mathjaxinline] <span style="color:#27408C"><b class="bf">factorial</b></span>. Expanding the factorial, we get the coefficient of the [mathjaxinline]\, y^ k\,[/mathjaxinline] to be </p><table id="a0000002110" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002111"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle {m \choose k}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{m!}{(m-k)!\, k!}[/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{m\, (m-1)\, (m-2)\, \cdots \, (m-k+2)\, (m-k+1)}{k\, (k-1)\, \cdots \, (3)\, (2)\, (1)}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.277)</td></tr></table><p>
This number is called the <span style="color:#27408C"><b class="bf">binomial coefficient</b></span>, and [mathjaxinline]\, \, \displaystyle {m \choose k}\, \,[/mathjaxinline] is read as “<span style="color:#27408C"><b class="bf">[mathjaxinline]m[/mathjaxinline] choose [mathjaxinline]k[/mathjaxinline]</b></span>." <br/></p><p><b class="bfseries">General formula for integral with one positive odd exponent of sine or cosine</b></p><p>
Let [mathjaxinline]\, 2m+1\,[/mathjaxinline] be any positive integer and [mathjaxinline]\, n\,[/mathjaxinline] any real number such that [mathjaxinline]\, n\neq -(2k+1)\,[/mathjaxinline] for any integer [mathjaxinline]\, k\,[/mathjaxinline] such that[mathjaxinline]\, 0\leq k\leq m,\, \,[/mathjaxinline] we obtain the following formulas:<br/></p><table id="a0000002112" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002113"><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 \sum _{k=0}^ m (-1)^ k\, {m \choose k} \, \frac{\sin ^{2k+n+1}(x)}{2k+n+1}\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.278)</td></tr><tr id="a0000002114"><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 \sum _{k=0}^ m (-1)^{k+1} \, {m \choose k} \, \frac{\cos ^{2k+n+1}(x)}{2k+n+1}\, +\, C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.279)</td></tr></table><p>
where </p><table id="a0000002115" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002116"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle {m \choose k}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{m!}{(m-k)!\, k!}[/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{m(m-1)(m-2)\cdots (m-k+2)(m-k+1)}{k(k-1)\cdots (3)(2)(1)}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.280)</td></tr></table><p>
are the <span style="color:#27408C"><b class="bf">binomial coefficients</b></span>.</p><p><b class="bfseries">General formula for integral with even positive power of secant</b></p><table id="a0000002117" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002118"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sec ^{2m}(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 \sec ^2(x) \left(1+\tan ^2(x)\right)^{m-1}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.281)</td></tr><tr id="a0000002119"><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 ^2(x)\left(\sum _{k=0}^{m-1} {m-1 \choose k} \tan ^{2k}(x)\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.282)</td></tr><tr id="a0000002120"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \int \sec ^{2m}(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 \int \left(\sum _{k=0}^{m-1} {m-1 \choose k} u^{2k}\right)\, du\qquad (u=\tan (x))[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.283)</td></tr><tr id="a0000002121"><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 \sum _{k=0}^{m-1} {m-1 \choose k} \frac{\tan ^{2k+1}(x)}{2k+1}+C[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.284)</td></tr></table>
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