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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Partial fractions</h3>
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<h2 class="hd hd-2 unit-title">2. Partial fractions</h2>
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<p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Use <span style="color:#27408C"><b class="bf">partial fractions</b></span> to rearrange rational functions into simpler pieces. </p></li><li><p>
Apply techniques of integration as needed to <span style="color:#27408C"><b class="bf">integrate rational functions</b></span>. </p></li></ul><p><h3>Contents: 12 pages</h3></p><p>
10 videos (84 minutes at 1x speed) 25 problems </p>
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<h2 class="hd hd-2 unit-title">3. Rational functions</h2>
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<h3 class="hd hd-2">Rational functions</h3>
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<p><span style="color:#27408C"><b class="bf">Polynomials</b></span> are functions [mathjaxinline]P(x) = a_ n x^ n + a_{n-1}x^{n-1} + \dotsb + a_1 x + a_0[/mathjaxinline] where the constants [mathjaxinline]a_ i[/mathjaxinline] are real numbers and the powers [mathjaxinline]n[/mathjaxinline] are non-negative, finite integers. </p><p>
The <span style="color:#27408C"><b class="bf">degree</b></span> of a polynomial [mathjaxinline]P(x)[/mathjaxinline] is the integer [mathjaxinline]\mathrm{deg}(P)=n[/mathjaxinline] of the highest order [mathjaxinline]x[/mathjaxinline] term in the polynomial [mathjaxinline]P(x)[/mathjaxinline]. For example, if [mathjaxinline]\, P(x)=-x^5-4x+1\,[/mathjaxinline] then [mathjaxinline]\mathrm{deg}(P)= 5.[/mathjaxinline]<br/></p><p><span style="color:#27408C"><b class="bf">Rational functions</b></span> are ratios of polynomials. For example, if [mathjaxinline]P(x)[/mathjaxinline] and [mathjaxinline]Q(x)[/mathjaxinline] are polynomials, then </p><table id="a0000002242" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
is a rational function. </p>
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Recognizing rational functions
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Which of the following functions are rational functions? </p>
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<text>[mathjaxinline]x^{-1}[/mathjaxinline]</text>
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<input type="checkbox" name="input_technique3-tab3-problem1_2_1[]" id="input_technique3-tab3-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="technique3-tab3-problem1_2_1-choice_3-label" for="input_technique3-tab3-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_technique3-tab3-problem1_2_1">
<text>[mathjaxinline]x^{2/3}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{x^2-1}{x^2+1}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \frac{x^2-1}{\sqrt {x^2+1}}[/mathjaxinline]</text>
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<input type="checkbox" name="input_technique3-tab3-problem1_2_1[]" id="input_technique3-tab3-problem1_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="technique3-tab3-problem1_2_1-choice_6-label" for="input_technique3-tab3-problem1_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_technique3-tab3-problem1_2_1">
<text>[mathjaxinline]\displaystyle e^ x+1[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">4. Method of partial fractions</h2>
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<p>
Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Step</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x^2+x-2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(1) Factor the denominator [mathjaxinline]Q(x)[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]Q(x) = x^2+x-2 = (x-1)(x+2)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Set-up </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{(x-1)(x+2)} = \frac{A}{x-1}+\frac{B}{x+2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) Solve for [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] </td><td style="text-align:left; border:none"><b class="bf">For </b>[mathjaxinline]A[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Multiply by [mathjaxinline](x-1)[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x+2} = A + \frac{B}{x+2}(x-1)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Plug in [mathjaxinline]x=1[/mathjaxinline]: (Take the limits as [mathjaxinline]x \rightarrow 1[/mathjaxinline].)</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4-1}{1+2} = A[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none"><b class="bf">For</b> [mathjaxinline]B[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Multiply by [mathjaxinline](x+2)[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x-1} = \frac{A}{x-1}(x+2) +B[/mathjaxinline] </td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Plug in [mathjaxinline]x=-2[/mathjaxinline]: (Take the limits as [mathjaxinline]x \rightarrow -2[/mathjaxinline].)</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{-8-1}{-2-1} = B[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(4) Rewrite rational function. </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x^2+x-2} = \frac{1}{x-1}+\frac{3}{x+2}[/mathjaxinline] </td></tr></table>
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Partial fractions practice step 1
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Factor the denominator </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{x-3}{x^2+x-6}[/mathjax]</td>
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Partial fractions practice step 3
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Use the method described above to write the rational function in the form: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{x-3}{x^2-3x-4} = \frac{A}{x-4} + \frac{B}{x+1}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(4.367)</td>
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(Enter as fractions.) </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]B=[/mathjaxinline]</p>
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Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]Q(x)[/mathjaxinline] can be factored linearly with distinct factors, and the degree of [mathjaxinline]P(x)[/mathjaxinline] is less than the degree of [mathjaxinline]Q(x)[/mathjaxinline], then the fastest way to apply the method of partial fractions is the following: </p><ol class="enumerate"><li value="1"><p>
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Use the method of cover-up to find the coefficients [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{x^2+3x+8}{(x-1)(x-2)(x+5)} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x+5}[/mathjaxinline] </p>
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Use the method of cover-up to find the coefficients [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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Use the method of cover-up to find the constants [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline].: </p>
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[mathjaxinline]\displaystyle \frac{x}{(x-2)(x+3)} = \frac{A}{x-2}+\frac{B}{x+3}[/mathjaxinline] </p>
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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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Practice integrating 1
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Use the computation above to evaluate the following integral. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{x}{(x-2)(x+3)}\, dx=[/mathjaxinline]</p>
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<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>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<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 integrating 2
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Evaluate the following integral. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{3x^2+4x-11}{(x^2-1)(x-2)}dx=[/mathjaxinline]</p>
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<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
<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. Repeated linear factors in denominator</h2>
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<p>
Given [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]\mathrm{deg}P < \mathrm{deg}Q[/mathjaxinline]. If [mathjaxinline]Q(x) = (x-a)^ n[/mathjaxinline], then the set-up for partial fractions is as follows: </p><table id="a0000002301" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)} = \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \frac{A_3}{(x-a)^3} + \dotsb + \frac{A_ n}{(x-a)^ n}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Repeated linear factors 1
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Find the constants [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{x^2}{(x+1)^2(x-1)} = \frac{A}{x-1}+\frac{B}{x+1} + \frac{C}{(x+1)^2}[/mathjaxinline] </p>
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(Enter as fractions.) </p>
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Repeated linear factors 2
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Find the constants [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and [mathjaxinline]D[/mathjaxinline] such that </p>
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[mathjaxinline]\displaystyle \frac{1}{(x+1)^2x^2} = \frac{A}{x}+\frac{B}{x^2} + \frac{C}{x+1} + \frac{D}{(x+1)^2}[/mathjaxinline] </p>
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(Enter as fractions.) </p>
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<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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Repeated linear factors 3
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Find the constants [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{3x+2}{x(x+1)^2} = \frac{A}{x}+\frac{B}{x+1} + \frac{C}{(x+1)^2}[/mathjaxinline] </p>
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(Enter as fractions.) </p>
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Repeated linear factors integral
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Use the computation above to evaluate the following integral. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{3x+2}{x(x+1)^2}dx=[/mathjaxinline]</p>
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<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<p>
Note that all polynomials with real coefficients can be factored into a product of linear and quadratic factors, where the quadratic factors cannot be factored further. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Method of partial fractions</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Factor denominator [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{x^2}{(x-1)(x^2+1)}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Setup </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{x^2}{(x-1)(x^2+1)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+1}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) Solve for [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]. </td><td style="text-align:left; border:none">
Cover-up [mathjaxinline]x-1[/mathjaxinline] to find [mathjaxinline]A[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{1^2+1} = \frac12 = A[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Clear denominator:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]x^2 = (1/2)(x^2+1) + (Bx+C)(x-1) = (1/2 + B)x^2 + (C-B)x+(1/2-C)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Solve system for [mathjaxinline]B[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline] looking at the coefficients of [mathjaxinline]x^2[/mathjaxinline] and [mathjaxinline]x^0[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]1 = 1/2 + B \quad \Longrightarrow \quad B = 1/2[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]0 = 1/2 - C \quad \Longrightarrow \quad C = 1/2[/mathjaxinline] </td></tr></table>
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Quadratic factors 1
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Find the constants [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{1}{x^3+1} = \frac{1}{(x+1)(x^2-x+1)}= \frac{A}{x+1}+\frac{Bx+C}{x^2-x+1}.[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]B=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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Quadratic factors 2
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Find the constants [mathjaxinline]A_1[/mathjaxinline], [mathjaxinline]B_1[/mathjaxinline], [mathjaxinline]A_2[/mathjaxinline], and [mathjaxinline]B_2[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{1}{(x^2+1)(x^2+2)} = \frac{A_1x+B_1}{x^2+1}+\frac{A_2x+B_2}{x^2+2}.[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_1=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]B_1=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]A_2=[/mathjaxinline]</p>
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Quadratic factors 3
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Find the constants [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]: </p>
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[mathjaxinline]\displaystyle \frac{2x-9}{(x^2+9)(x+2)} = \frac{Ax+B}{x^2+9}+\frac{C}{x+2}[/mathjaxinline] </p>
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(Enter as fractions.) </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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Quadratic factors integral
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Use your computation above to evaluate the following integral. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{2x-9}{(x^2+9)(x+2)}dx=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">9. Integrating rational functions</h2>
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<h3 class="hd hd-2">Integrating rational functions</h3>
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Find the integral 1
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Find the integral of [mathjaxinline]\displaystyle \int \frac{dx}{x^4-x^2}[/mathjaxinline]. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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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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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<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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Find the integral 2
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Find the integral of [mathjaxinline]\displaystyle \int \frac{dx}{x^4+x^2}[/mathjaxinline]. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<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">
<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 \( \sqrt{x^2-y} \)
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">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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Find the integral 3
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Find the integral of [mathjaxinline]\displaystyle \int \frac{2}{3x-1}\, dx[/mathjaxinline]. </p>
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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">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">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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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">10. Improper fractions</h2>
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<p>
To integrate a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]\mathrm{deg}P \geq \mathrm{deg}Q[/mathjaxinline]: </p><ol class="enumerate"><li value="1"><p>
Use Long division to divide [mathjaxinline]Q(x)[/mathjaxinline] into [mathjaxinline]P(x)[/mathjaxinline] to obtain </p><table id="a0000002371" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)} = p(x) + \frac{r(x)}{Q(x)},[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]p(x)[/mathjaxinline] and [mathjaxinline]r(x)[/mathjaxinline] are polynomials, and [mathjaxinline]\mathrm{deg}r < \mathrm{deg}Q[/mathjaxinline]. </p></li><li value="2"><p>
Use the method of partial fractions to divide [mathjaxinline]\displaystyle \frac{r(x)}{Q(x)}[/mathjaxinline] into simpler pieces to integrate. </p></li></ol>
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Long division of polynomials 1
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Use long division on [mathjaxinline]\displaystyle \frac{x^2}{x^2-1}[/mathjaxinline] to find the quotient [mathjaxinline]q(x)[/mathjaxinline] and remainder [mathjaxinline]r(x)[/mathjaxinline] such that </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000002372" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{x^2}{x^2-1}= q(x) + \frac{r(x)}{x^2-1}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p style="display:inline">[mathjaxinline]q(x) =[/mathjaxinline]</p>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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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>
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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>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Long division of polynomials 2
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Use long division on [mathjaxinline]\displaystyle \frac{x^3}{x^2-1}[/mathjaxinline] to find the quotient [mathjaxinline]q(x)[/mathjaxinline] and remainder [mathjaxinline]r(x)[/mathjaxinline] such that </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000002378" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{x^3}{x^2-1}= q(x) + \frac{r(x)}{x^2-1}.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p style="display:inline">[mathjaxinline]q(x) =[/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>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
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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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Long division of polynomials 3
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Use long division on [mathjaxinline]\displaystyle \frac{x^8}{(x+2)^2(x-2)^2}[/mathjaxinline] to find the quotient [mathjaxinline]q(x)[/mathjaxinline] and remainder [mathjaxinline]r(x)[/mathjaxinline] such that </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{x^8}{(x+2)^2(x-2)^2} = q(x) + \frac{r(x)}{(x+2)^2(x-2)^2}.[/mathjax]</td>
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<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">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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Find the integral 1 (*)
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Use your answer from part above to integrate the function. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{x^8}{(x+2)^2(x-2)^2}dx=[/mathjaxinline]</p>
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<font color="#0078b0">2520</font>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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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>
</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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Find the integral 2
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Find the integral of [mathjaxinline]\displaystyle \int \frac{x+2}{3x-1}\, dx[/mathjaxinline]. </p>
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(Enter [mathjaxinline]C[/mathjaxinline] for the constant of integration.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int \frac{x+2}{3x-1}dx=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">11. General pattern</h2>
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Identify the set-up
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Identify the setup for [mathjaxinline]\displaystyle \frac{x^3-3}{(x^2+1)^2}[/mathjaxinline]. </p>
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<text> [mathjaxinline]\displaystyle \frac{A_1}{x^2+1} + \frac{A_2}{(x^2+1)^2}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{A_1+ B_1x}{x^2+1} + \frac{A_2 + B_2x}{(x^2+1)^2}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \frac{A_1+ B_1x}{x^2+1} + \frac{A_2 + B_2x+C_2 x^2 + D_2 x^3}{(x^2+1)^2}[/mathjaxinline]</text>
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<h3 class="hd hd-2">General pattern</h3>
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<h1>Summary</h1><p><b class="bfseries">Rational functions</b></p><p><span style="color:#27408C"><b class="bf">Polynomials</b></span> are functions [mathjaxinline]P(x) = a_ n x^ n + a_{n-1}x^{n-1} + \dotsb + a_1 x + a_0[/mathjaxinline] where the constants [mathjaxinline]a_ i[/mathjaxinline] are real numbers and the powers [mathjaxinline]n[/mathjaxinline] are non-negative, finite integers. </p><p>
The <span style="color:#27408C"><b class="bf">degree</b></span> of a polynomial [mathjaxinline]P(x)[/mathjaxinline] is the integer [mathjaxinline]\mathrm{deg}(P)=n[/mathjaxinline] of the highest order [mathjaxinline]x[/mathjaxinline] term in the polynomial [mathjaxinline]P(x)[/mathjaxinline]. </p><p><span style="color:#27408C"><b class="bf">Rational functions</b></span> are ratios of polynomials. For example, if [mathjaxinline]P(x)[/mathjaxinline] and [mathjaxinline]Q(x)[/mathjaxinline] are polynomials, then </p><table id="a0000002407" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
is a rational function. </p><p><b class="bfseries">Method of partial fractions</b></p><p>
Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Step</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x^2+x-2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(1) Factor the denominator [mathjaxinline]Q(x)[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]Q(x) = x^2+x-2 = (x-1)(x+2)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Set-up </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{(x-1)(x+2)} = \frac{A}{x-1}+\frac{B}{x+2}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) Solve for [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] </td><td style="text-align:left; border:none">
Multiply by [mathjaxinline](x-1)[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x+2} = A + \frac{B}{x+2}(x-1)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Plug in [mathjaxinline]x=1[/mathjaxinline]: (Take the limits as [mathjaxinline]x \rightarrow 1[/mathjaxinline].)</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4-1}{1+2} = A[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Multiply by [mathjaxinline](x+2)[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x-1} = \frac{A}{x-1}(x+2) +B[/mathjaxinline] </td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Plug in [mathjaxinline]x=-2[/mathjaxinline]: (Take the limits as [mathjaxinline]x \rightarrow -2[/mathjaxinline].)</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{-8-1}{-2-1} = B[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(4) Rewrite rational function. </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{4x-1}{x^2+x-2} = \frac{1}{x-1}+\frac{3}{x+2}[/mathjaxinline] </td></tr></table><p><b class="bfseries">Cover-up method</b></p><p>
Given a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]Q(x)[/mathjaxinline] can be factored linearly with distinct factors, and the degree of [mathjaxinline]P(x)[/mathjaxinline] is less than the degree of [mathjaxinline]Q(x)[/mathjaxinline], then the fastest way to apply the method of partial fractions is the following: </p><ol class="enumerate"><li value="1"><p>
Factor [mathjaxinline]Q(x)[/mathjaxinline] </p></li><li value="2"><p>
Set-up </p></li><li value="3"><p>
Cover-up </p></li></ol><p><b class="bfseries">Repeated linear factors</b></p><p>
Given [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]\mathrm{deg}P < \mathrm{deg}Q[/mathjaxinline]. If [mathjaxinline]Q(x) = (x-a)^ n[/mathjaxinline], then the set-up for partial fractions is as follows: </p><table id="a0000002408" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)} = \frac{A_1}{(x-a)} + \frac{A_2}{(x-a)^2} + \frac{A_3}{(x-a)^3} + \dotsb + \frac{A_ n}{(x-a)^ n}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Quadratic factors in the denominator</b></p><p>
Note that all polynomials with real coefficients can be factors into a product of linear and quadratic factors, where the quadratic factors cannot be factored further. </p><table class="tabular" cellspacing="0" style="table-layout:auto"><tr><td style="text-align:left; border:none"><b class="bf">Method of partial fractions</b></td><td style="text-align:left; border:none"><b class="bf">Example</b></td></tr><tr><td style="text-align:left; border:none">
(1) Factor denominator [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{x^2}{(x-1)(x^2+1)}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(2) Setup </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{x^2}{(x-1)(x^2+1)} = \frac{A}{x-1} + \frac{Bx+C}{x^2+1}[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none">
(3) Solve for [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline]. </td><td style="text-align:left; border:none">
Cover-up [mathjaxinline]x-1[/mathjaxinline] to find [mathjaxinline]A[/mathjaxinline]:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{1^1+1} = \frac12 = A[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Clear denominator:</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]x^2 = (1/2)(x^2+1) + (Bx+C)(x-1) = (1/2 + B)x^2 + (C-B)x+(1/2-C)[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
Solve system for [mathjaxinline]B[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline] looking at the coefficients of [mathjaxinline]x^2[/mathjaxinline] and [mathjaxinline]x^0[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]1 = 1/2 + B \quad \Longrightarrow \quad B = 1/2[/mathjaxinline]</td></tr><tr><td style="text-align:left; border:none"> </td><td style="text-align:left; border:none">
[mathjaxinline]0 = 1/2 - C \quad \Longrightarrow \quad C = 1/2[/mathjaxinline] </td></tr></table><p><b class="bfseries">Improper fractions</b></p><p>
To integrate a rational function [mathjaxinline]\displaystyle \frac{P(x)}{Q(x)}[/mathjaxinline] such that [mathjaxinline]\mathrm{deg}P \geq \mathrm{deg}Q[/mathjaxinline]: </p><ol class="enumerate"><li value="1"><p>
Use Long division to divide [mathjaxinline]Q(x)[/mathjaxinline] into [mathjaxinline]P(x)[/mathjaxinline] to obtain </p><table id="a0000002409" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{P(x)}{Q(x)} = p(x) + \frac{r(x)}{Q(x)},[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
where [mathjaxinline]p(x)[/mathjaxinline] and [mathjaxinline]r(x)[/mathjaxinline] are polynomials, and [mathjaxinline]\mathrm{deg}r < \mathrm{deg}Q[/mathjaxinline]. </p></li><li value="2"><p>
Use the method of partial fractions to divide [mathjaxinline]\displaystyle \frac{r(x)}{Q(x)}[/mathjaxinline] into simpler pieces to integrate. </p></li></ol>
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