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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Improper integrals</h3>
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<h2 class="hd hd-2 unit-title">2. Improper integrals</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Apply l'Hôpital's rule to evaluate <span style="color:#27408C"><b class="bf">improper integrals</b></span>. </p></li><li><p>
Apply <span style="color:#27408C"><b class="bf">limit comparison</b></span> and <span style="color:#27408C"><b class="bf">comparison</b></span> to known improper integrals to determine if an improper integral <span style="color:#27408C"><b class="bf">converges</b></span> or <span style="color:#27408C"><b class="bf">diverges</b></span>. </p></li><li><p>
Use <span style="color:#27408C"><b class="bf">limiting methods</b></span> to handle integrals where the integrand tends to infinity as [mathjaxinline]x[/mathjaxinline] approaches a finite value. </p></li></ul><p><b class="bfseries">Contents: 16 pages</b></p><p>
12 videos (93 minutes 1x speed) 24 questions </p>
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<h2 class="hd hd-2 unit-title">3. Review</h2>
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<h3 class="hd hd-2">Review of L'Hospital's rule and growth rates</h3>
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<p><b class="bfseries">Review of l'Hôpital's rule</b></p><p>
Here we recall l'Hôpital's rule in one case, [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \begin{cases} f(x) \longrightarrow \infty \\ g(x) \longrightarrow \infty \\ \displaystyle \frac{f'}{g'} \longrightarrow L \end{cases}[/mathjaxinline] as [mathjaxinline]x \rightarrow a[/mathjaxinline], </p><p>
then [mathjaxinline]\displaystyle \frac{f}{g} \longrightarrow L[/mathjaxinline] as [mathjaxinline]x \rightarrow a[/mathjaxinline]. </p><p>
(Recall that [mathjaxinline]a = \pm \infty[/mathjaxinline] and [mathjaxinline]L = \pm \infty[/mathjaxinline] is OK.) </p><p><b class="bfseries">Rate of Growth (as [mathjaxinline]x\rightarrow \infty[/mathjaxinline])</b></p><p>
Consider [mathjaxinline]\ f, g >0[/mathjaxinline]. We say that [mathjaxinline]g[/mathjaxinline] grows faster than [mathjaxinline]\ f[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline], and write this as </p><table id="a0000000847" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 0< f(x) << g(x),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
if [mathjaxinline]\displaystyle \ f,g \longrightarrow \infty[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{f(x)}{g(x)} \longrightarrow 0[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><p><b class="bfseries">Examples</b></p><p>
For [mathjaxinline]p > 0,[/mathjaxinline] </p><p>
[mathjaxinline]\displaystyle \ln x << x^ p << e^ x << e^{x^2}.[/mathjaxinline] </p><p>
Note that [mathjaxinline]e^{x^2} = e^{(x^2)}[/mathjaxinline], not [mathjaxinline](e^ x)^2 = e^{2x}[/mathjaxinline]. </p><p><b class="bfseries">Rate of Decay (as [mathjaxinline]x\rightarrow \infty[/mathjaxinline])</b></p><p>
Consider [mathjaxinline]\ f, g >0[/mathjaxinline]. We say that [mathjaxinline]\ f[/mathjaxinline] decays faster than [mathjaxinline]g[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline], and write this as </p><table id="a0000000848" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle g(x) >> f(x)>0,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
if [mathjaxinline]\displaystyle \ f,g \longrightarrow 0[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}\longrightarrow 0[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><p><b class="bfseries">Examples</b></p><p>
For [mathjaxinline]p >0,[/mathjaxinline] </p><table id="a0000000849" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{1}{\ln x} >> \frac{1}{x^ p} >> e^{-x} >> e^{-x^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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Practice with rates of decay
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Suppose that [mathjaxinline]h(x)&gt;0[/mathjaxinline] and [mathjaxinline]g(x)&gt;0[/mathjaxinline] both tend towards [mathjaxinline]0[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline]. We know that [mathjaxinline]g(x)[/mathjaxinline] decays faster than [mathjaxinline]h(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline]. That is [mathjaxinline]h(x)&gt;&gt; g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]. </p>
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Which of the following functions also decay faster than [mathjaxinline]h(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]? (Choose all that apply.) </p>
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<h2 class="hd hd-2 unit-title">4. Improper integrals</h2>
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<p><b class="bfseries">Improper integrals definition</b></p><p>
An <span style="color:#27408C"><b class="bf">improper integral</b></span> is defined by [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx = \lim _{N\rightarrow \infty } \int _ a^{N} f(x) \, dx.[/mathjaxinline] </p><p>
This improper integral <span style="color:#27408C"><b class="bf">converges</b></span> if the limit exists and is finite.<br/>This improper integral <span style="color:#27408C"><b class="bf">diverges</b></span> if the limit does not exist (this includes when the limit is [mathjaxinline]\pm \infty[/mathjaxinline]). </p>
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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(If divergent, enter <b class="bf">div</b>. Note you can also enter DIV or Div the grader is not case sensitive.) </p>
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<p style="display:inline">For [mathjaxinline]p&gt;1[/mathjaxinline], [mathjaxinline]\displaystyle \int _1^{\infty }x^{-p}\, dx=[/mathjaxinline]</p>
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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<p style="display:inline">For [mathjaxinline]0&lt;p &lt; 1[/mathjaxinline], [mathjaxinline]\displaystyle \int _1^{\infty }x^{-p}\, dx=[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</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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<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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<h2 class="hd hd-2 unit-title">5. Applications to probability</h2>
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To review probability, you may want to revisit Calculus 1B / Unit 3: Applications / Probability.</p>
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Probability distribution
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<p>
Recall that the <span style="color:#27408C"><b class="bf">exponential distribution</b></span> is used to compute the probability of a &#8220;waiting time" between events, such as the arrival of a bus, the decay of a radioactive particle, or the time between phone calls during business hours. </p>
<p>
Recall that the <span style="color:#27408C"><b class="bf">exponential probability distribution</b></span> takes the form [mathjaxinline]p(t) = A e^{-kt}[/mathjaxinline] where [mathjaxinline]k&gt;0[/mathjaxinline], and must satisfy [mathjaxinline]\displaystyle \int _0^{\infty } A e^{-k t} \, dt =1[/mathjaxinline]. </p>
<p>
Find the value of [mathjaxinline]A[/mathjaxinline] in terms of [mathjaxinline]k[/mathjaxinline] such that the improper integral evaluates to [mathjaxinline]1[/mathjaxinline], in other words, find the [mathjaxinline]A[/mathjaxinline] that normalizes the distribution [mathjaxinline]e^{-kt}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]A=[/mathjaxinline]</p>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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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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Expected value
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Recall that the expected value [mathjaxinline]E[/mathjaxinline] of an exponential distribution [mathjaxinline]p(t)[/mathjaxinline] is given by </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000862" style="table-layout:auto" width="100%">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle E = \int _0^{\infty } t \, p(t) \, dt[/mathjax]</td>
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where [mathjaxinline]E[/mathjaxinline] is the expected value, and [mathjaxinline]p(t)[/mathjaxinline] is the probability distribution. </p>
<p>
Suppose that at Kendall Station near MIT, the expected wait time between trains is 8 minutes, and the distribution for time between trains is exponential. </p>
<p>
Find the exponential distribution [mathjaxinline]p(t) = Ae^{-kt}[/mathjaxinline] where [mathjaxinline]t[/mathjaxinline] is measured in minutes that models the wait time between trains at Kendall station. In other words, solve for [mathjaxinline]\, k\,[/mathjaxinline] and find the function [mathjaxinline]\, p(t)\,[/mathjaxinline] in terms of [mathjaxinline]t[/mathjaxinline] only.<br/></p>
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<p style="display:inline">[mathjaxinline]p(t)=[/mathjaxinline]</p>
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Probability problem 3
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Recall that the <b class="bf"><span style="color:#27408C">standard deviation</span></b> [mathjaxinline]\sigma[/mathjaxinline] is the (positive) square root of the <b class="bf"><span style="color:#27408C">variance</span></b> [mathjaxinline]\, \displaystyle \sigma ^2,\,[/mathjaxinline] which is defined as </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000879" style="table-layout:auto" width="100%">
<tr id="a0000000880">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \sigma ^2 = \int _0^{\infty } (t-E)^2p(t) \, dt \qquad (\text {variance}).[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(2.68)</td>
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<p>
In this formula [mathjaxinline]E[/mathjaxinline] is the expected value, and [mathjaxinline]p(t)[/mathjaxinline] is the probability distribution. </p>
<p>
Suppose that at Kendall Station near MIT, the expected wait time between trains is 8 minutes, and the distribution for time between trains is exponential. </p>
<p>
Use integration by parts to compute the standard deviation [mathjaxinline]\sigma[/mathjaxinline] of the wait times at Kendall station. </p>
<p>
<p style="display:inline">[mathjaxinline]\sigma =[/mathjaxinline]</p>
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<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">6. First divergent example</h2>
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<p><b class="bfseries">Conclusion of example: powers of [mathjaxinline]x[/mathjaxinline]</b></p><p>
For [mathjaxinline]a>0[/mathjaxinline]: [mathjaxinline]\displaystyle \int _ a^{\infty } \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if } \, \, p\leq 1\\ \text {converges to} \, \, \frac{a^{-p+1}}{p-1} & \text {if }\, \, p> 1\end{cases}[/mathjaxinline] </p>
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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(If divergent, enter <b class="bf">div</b>.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{10}^{\infty } x^{-1/3} \, dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _2^{\infty } x^{-3} \, dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _5^{\infty } x^3 \, dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{-\infty }^{-5} x^3 \, dx=[/mathjaxinline]</p>
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<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
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<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<td class="formulainput">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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Practice 1
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
<p>
(If divergent, enter <b class="bf">div</b>.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \int _ e^{\infty }\frac{dx}{x(\ln x)^2}=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
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<th class="formulainput" scope="col">Example Entries</th>
</tr>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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</tr>
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<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>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Practice 2
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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(If divergent, enter <b class="bf">div</b>.) <p style="display:inline">[mathjaxinline]\displaystyle \int _ e^{\infty }\frac{dx}{x(\ln x)}=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_improper-tab7-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</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>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Practice 3
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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(If divergent, enter <b class="bf">div</b>.) </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \int _0^{\infty }\frac{dx}{(x+2)^3}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">8. Limit comparison</h2>
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<p><b class="bfseries">Notation</b></p><p>
We say that [mathjaxinline]f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline] if [mathjaxinline]\displaystyle \frac{f(x)}{g(x)} \underset {x \rightarrow \infty }{\longrightarrow } 1[/mathjaxinline]. </p><p>
In words, we say that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] are <span style="color:#27408C"><b class="bf">similar</b></span> as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]. (The idea is that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] have the same asymptotic behavior as [mathjaxinline]x[/mathjaxinline] tends to infinity.) </p><p><b class="bfseries">Limit comparison</b></p><p>
The idea behind limit comparison is that if the asymptotic behavior is the same, then the improper integrals have the same behavior. </p><p>
Let [mathjaxinline]\, \displaystyle f(x), g(x)\geq 0[/mathjaxinline].<br/></p><p>
If [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline], <br/>then the two integrals [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] and [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] (for large [mathjaxinline]a[/mathjaxinline]) either <span style="color:#27408C"><b class="bf">both converge</b></span> or <span style="color:#27408C"><b class="bf">both diverge</b></span>. </p><p>
This also works in the case where one function decays faster than the other as [mathjaxinline]x[/mathjaxinline] tends towards infinity. </p><p>
Suppose that [mathjaxinline]\ g(x)[/mathjaxinline] decays faster than [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]. That is [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><ul class="itemize"><li><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] converges. </p></li><li><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] diverges. </p></li></ul>
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Limit comparison practice
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Use limit comparison to an integral of the form [mathjaxinline]\displaystyle \int _ a^{\infty } \frac{dx}{x^ p}[/mathjaxinline] to determine if the following integrals converge or diverge. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{100}^{\infty } \frac{dx}{x^{42} - 1}[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{100}^{\infty } \frac{dx}{(x^{5} - 1)^{1/5}}[/mathjaxinline]</p>
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<option value="converges"> converges</option>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{100}^{\infty } \frac{dx}{(x^{1/2} - 1)^3}[/mathjaxinline]</p>
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<option value="converges"> converges</option>
<option value="diverges"> diverges</option>
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Decay rate and improper integrals
</h3>
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<p>
Let [mathjaxinline]\, \displaystyle f(x), g(x)\geq 0[/mathjaxinline].<br/></p>
<p>
Suppose [mathjaxinline]\ g(x)[/mathjaxinline] decays faster than [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends to infinity. That is [mathjaxinline]\ f(x) &gt;&gt; g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline] (and both [mathjaxinline]\ f, g \longrightarrow 0[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]). </p>
<p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] converges, which of the following must also converge? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_improper-tab8-problem2_2_1">
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \sqrt {f(x)} \, dx[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_improper-tab8-problem2_2_1[]" id="input_improper-tab8-problem2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="improper-tab8-problem2_2_1-choice_1-label" for="input_improper-tab8-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_improper-tab8-problem2_2_1">
<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \left(f(x)\right)^2 \, dx[/mathjaxinline]</text>
</label>
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \sqrt {g(x)} \, dx[/mathjaxinline]</text>
</label>
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_improper-tab8-problem2_2_1[]" id="input_improper-tab8-problem2_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="improper-tab8-problem2_2_1-choice_4-label" for="input_improper-tab8-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_improper-tab8-problem2_2_1">
<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \left(g(x)\right)^2 \, dx[/mathjaxinline]</text>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</p>
<p>
Suppose [mathjaxinline]\ g(x)[/mathjaxinline] decays faster than [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends to infinity. That is [mathjaxinline]\ f(x) &gt;&gt; g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline] (and both [mathjaxinline]\ f, g \longrightarrow 0[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]). </p>
<p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] diverges, which of the following must also diverge? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_improper-tab8-problem2_3_1">
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \sqrt {f(x)} \, dx[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_improper-tab8-problem2_3_1[]" id="input_improper-tab8-problem2_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="improper-tab8-problem2_3_1-choice_1-label" for="input_improper-tab8-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_improper-tab8-problem2_3_1">
<text>[mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_improper-tab8-problem2_3_1[]" id="input_improper-tab8-problem2_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="improper-tab8-problem2_3_1-choice_2-label" for="input_improper-tab8-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_improper-tab8-problem2_3_1">
<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \left(f(x)\right)^2 \, dx[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \sqrt {g(x)} \, dx[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _ a^{\infty } \left(g(x)\right)^2 \, dx[/mathjaxinline]</text>
</label>
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<h2 class="hd hd-2 unit-title">9. Decay rate and comparison of improper integrals</h2>
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<h3 class="hd hd-2">Comparison</h3>
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<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_improper-tab9-video1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_improper-tab9-video1">
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<p><b class="bfseries">Comparison</b></p><p>
Suppose [mathjaxinline]f(x) \geq g(x)>0[/mathjaxinline] for [mathjaxinline]x \geq a[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] converges also. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] diverges also. </p>
</div>
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<h2 class="hd hd-2 unit-title">10. Comparison practice</h2>
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<h3 class="hd hd-2">Apply comparison</h3>
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Comparison practice 1
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<p>
Recall that [mathjaxinline]\frac{1}{\ln x} &gt;&gt; \frac{1}{x^ p} &gt;&gt; e^{-x}[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p>
<p>
Apply comparison to one of the functions above to determine which of the following integrals converge and diverge. </p>
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[mathjaxinline]\displaystyle \int _{0}^{\infty } e^{-x}\sin {5x}\, dx[/mathjaxinline] </td>
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<option value="converges"> converges</option>
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<p><i class="itshape">Hint:</i> Compare the integral first with [mathjaxinline]\, \displaystyle \int _{0}^{\infty }\left| e^{-x}\sin {5x}\right|\, dx[/mathjaxinline].<br/></p>
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Comparison practice 2
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<p>
Recall that [mathjaxinline]\frac{1}{\ln x} &gt;&gt; \frac{1}{x^ p} &gt;&gt; e^{-x}[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p>
<p>
Apply comparison to one of the functions above to determine which of the following integrals converge and diverge. </p>
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[mathjaxinline]\displaystyle \int _{2}^{\infty } \frac{x^3}{\ln {x}} \, dx[/mathjaxinline] </td>
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<option value="converges"> converges</option>
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[mathjaxinline]\displaystyle \int _{1}^{\infty } \frac{\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx[/mathjaxinline] </td>
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Comparison practice 3
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Recall that [mathjaxinline]\frac{1}{\ln x} &gt;&gt; \frac{1}{x^ p} &gt;&gt; e^{-x}[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p>
<p>
Apply comparison to one of the functions above to determine which of the following integrals converge and diverge. </p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
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<td style="text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{1}^{\infty } \frac{e^{-x}}{x} \, dx[/mathjaxinline] </td>
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<option value="converges"> converges</option>
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[mathjaxinline]\displaystyle \int _{1}^{\infty } \frac{\sin ^2{5x}}{1+x^2} \, dx[/mathjaxinline] </td>
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<option value="converges"> converges</option>
<option value="diverges"> diverges</option>
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<h2 class="hd hd-2 unit-title">11. Singularities</h2>
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<p><b class="bfseries">Definition of singularity</b></p><p>
A <span style="color:#27408C"><b class="bf">singularity</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] is a point [mathjaxinline]x=s[/mathjaxinline] such that the function [mathjaxinline]\ f(x)[/mathjaxinline] does not exist at [mathjaxinline]x=s.[/mathjaxinline] </p><p>
There are three main ways that the function can fail to exist at a point: </p><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{+}} \left| f(x)\right|[/mathjaxinline] and/or [mathjaxinline]\displaystyle \lim _{x \rightarrow s^{-}} \left|f(x)\right|[/mathjaxinline] tends to [mathjaxinline]\infty[/mathjaxinline]. This is the case of most interest in this section. </p></li><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{\pm }} f(x)[/mathjaxinline] does not exist. In this case the function [mathjaxinline]\ f[/mathjaxinline] may oscillate, or have a jump discontinuity. </p></li><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{\pm }} f(x)[/mathjaxinline] exists and is finite. In this case, the function [mathjaxinline]\ f[/mathjaxinline] has a removable discontinuity. </p></li></ul><p>
(Note continuity was discussed in Calculus 1A: Differentiation in Unit 0: Limits.) </p>
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Review of singularities
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For each of the functions below, find its (finite) singularities. </p>
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(If there is more than one singularity, separate multiple answers with a comma: e.g. 0, 1, pi. If there are no singularities, type none.) </p>
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[mathjaxinline]\displaystyle \frac{1}{\sqrt {x^2+5}}[/mathjaxinline] </td>
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[mathjaxinline]\displaystyle \frac{1}{x^2+x}[/mathjaxinline] </td>
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[mathjaxinline]\displaystyle \frac{e^{-x}}{x}[/mathjaxinline]</td>
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[mathjaxinline]\displaystyle \frac{\ln |x|}{x^2-1}[/mathjaxinline]</td>
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<h3 class="hd hd-2">Improper integrals of second type</h3>
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<p><b class="bfseries">Definition of improper integrals of the 2nd type</b></p><p>
An <span style="color:#27408C"><b class="bf">improper integral of the 2nd type</b></span> is an integral [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] such that the function [mathjaxinline]\ f(x)[/mathjaxinline] has a singularity at [mathjaxinline]x=s[/mathjaxinline] for some [mathjaxinline]s[/mathjaxinline] with [mathjaxinline]a \leq s \leq b[/mathjaxinline]. </p><p>
For example, if [mathjaxinline]f(x)[/mathjaxinline] has a singularity at [mathjaxinline]x=b[/mathjaxinline], </p><p>
then [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx = \lim _{C \rightarrow b^{-}} \int _ a^ C f(x) \, dx.[/mathjaxinline] </p><p>
We say </p><ul class="itemize"><li><p>
the integral <span style="color:#27408C"><b class="bf">converges</b></span> if the limit exists and is finite. </p></li><li><p>
the integral <span style="color:#27408C"><b class="bf">diverges</b></span> if the limit does not exist (which includes the case that the limit is [mathjaxinline]\pm \infty[/mathjaxinline].) </p></li></ul>
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Identify the improper integrals
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Choose all of the following integrals that are improper integrals <b class="bf">of the second type</b>. Note that whether or not an integral is improper of the 1st type doesn't affect whether or not it is improper of the 2nd type. </p>
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(Select all that apply.) </p>
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<text>[mathjaxinline]\displaystyle \int _1^{\infty } \frac{dx}{\sqrt {x^3+5}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _1^{\infty } \frac{x^{2}\, dx}{x^3-8}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{1}^{2}\frac{dx}{x^2+x}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{0}^{1}\frac{dx}{\sqrt {1-x^3}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{0}^{\infty }\frac{e^{-x}\, dx}{x}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{1}^{\infty }\frac{\ln (x) \, dx}{x^2}[/mathjaxinline]</text>
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Convergent and divergent powers of x
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Decide whether the integral [mathjaxinline]\displaystyle \int _0^1 x^{-p}\, dx[/mathjaxinline] is convergent or divergent for different values of [mathjaxinline]p[/mathjaxinline]. </p>
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(If convergent, evaluate the integral. If divergent, enter DIV.) </p>
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<p style="display:inline">For [mathjaxinline]p=1[/mathjaxinline], &#8201; [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x} =[/mathjaxinline]</p>
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<p style="display:inline">For [mathjaxinline]p&gt;1[/mathjaxinline],&#8201; [mathjaxinline]\displaystyle \int _0^1 x^{-p}\, dx =[/mathjaxinline]</p>
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<p style="display:inline">For [mathjaxinline]0&lt;p&lt;1[/mathjaxinline], &#8201; [mathjaxinline]\displaystyle \int _0^1 x^{-p}\, dx =[/mathjaxinline]</p>
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The improper integral [mathjaxinline]\displaystyle \int _ a^{\infty } \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if}\, \, p\leq 1\\ \text {converges to} \displaystyle \, \, \frac{1}{a^{p-1} \, (p-1)} & \text {if}\, \, p> 1\end{cases}[/mathjaxinline]. </p><p>
The improper integral [mathjaxinline]\displaystyle \int _0^{a} \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if}\, \, p\geq 1\\ \text {converges to}\displaystyle \, \, \frac{a^{1-p}}{1-p} & \text {if}\, \, p< 1\end{cases}[/mathjaxinline] </p><div id="a0000000927" class="figure"><center><img src="/assets/courseware/v1/70ddb1e9817d840ec98831c4785d7b02/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_improper-01.svg" width="550px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 33</b>: <span>From left to right, we see the areas for [mathjaxinline]0 \leq x \leq 1[/mathjaxinline] and [mathjaxinline]1 \leq x < \infty[/mathjaxinline]<br/>under the graphs of [mathjaxinline]\displaystyle \frac{1}{x}[/mathjaxinline], [mathjaxinline]\displaystyle \frac{1}{x^2}[/mathjaxinline], and [mathjaxinline]\displaystyle \frac{1}{\sqrt {x}}[/mathjaxinline].<br/>The areas shaded in pink are infinite. The areas shaded in green are finite.</span></div></center></div>
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<h2 class="hd hd-2 unit-title">14. Improper integral practice</h2>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^2\frac{x}{\sqrt {4-x^2}}dx=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _0^2\frac{dx}{\sqrt {2-x}}=[/mathjaxinline]</p>
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Decide whether the following integral is convergent or divergent. If convergent, evaluate the integral. </p>
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(If divergent, enter <b class="bf">div</b>.) </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{\pi /2} \frac{dx}{x\sin ^2 x}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">15. Comparison and limit comparison for improper integrals of type 2</h2>
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<p><b class="bfseries">Review of function comparison</b></p><p>
Suppose that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] both have a singularity at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
Suppose [mathjaxinline]f(x) \geq g(x)\geq 0[/mathjaxinline] for all [mathjaxinline]a\leq x \leq b[/mathjaxinline] except at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] converges also. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] diverges also. </p>
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<p><b class="bfseries">Notation for similarity near singularities:</b></p><p>
Suppose that [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]\ g(x)[/mathjaxinline] have a singularity at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
We say that [mathjaxinline]\ f(x)[/mathjaxinline] is <span style="color:#27408C"><b class="bf">similar</b></span> to [mathjaxinline]\ g(x)[/mathjaxinline], and write [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow s^{+}[/mathjaxinline] or [mathjaxinline]x\rightarrow s^{-}[/mathjaxinline] if </p><table id="a0000000933" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000934"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{f(x)}{g(x)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 1 \qquad \text { as } \quad x \longrightarrow s^{\pm }.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.99)</td></tr></table><p>
We say that [mathjaxinline]\ f(x)[/mathjaxinline] grows faster than [mathjaxinline]\ g(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]s[/mathjaxinline], and write [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow s^{\pm }[/mathjaxinline], if [mathjaxinline]\displaystyle \ \ \begin{cases} f(x) \longrightarrow \infty \\ g(x) \longrightarrow \infty \\ \frac{g(x)}{f(x)} \longrightarrow 0 \end{cases}[/mathjaxinline] as [mathjaxinline]\displaystyle x \longrightarrow s^{\pm }.[/mathjaxinline] </p>
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<p><b class="bfseries">Limit comparison tests for improper integrals of 2nd type</b></p><p>
Suppose that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] both have a singularity at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
Suppose [mathjaxinline]f(x), g(x)\geq 0[/mathjaxinline] for all [mathjaxinline]a\leq x \leq b[/mathjaxinline] except at [mathjaxinline]x=s[/mathjaxinline]. </p><ol class="enumerate"><li value="1"><p>
If [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow s^{\pm }[/mathjaxinline], <br/>then the two integrals [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] and [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] either <span style="color:#27408C"><b class="bf">both converge</b></span> or <span style="color:#27408C"><b class="bf">both diverge</b></span>. </p></li><li value="2"><p>
Suppose that [mathjaxinline]\ f(x)[/mathjaxinline] grows faster than [mathjaxinline]\ g(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]s^{\pm }[/mathjaxinline]. In other words, [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow s^{\pm }[/mathjaxinline]. </p><ul class="itemize"><li><p>
If [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] converges. </p></li><li><p>
If [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] diverges. </p></li></ul></li></ol><p>
Note that this notation is exactly the same notation that we had before. The only difference is that instead of having [mathjaxinline]x \rightarrow \infty[/mathjaxinline], we have [mathjaxinline]x \rightarrow s^{+}[/mathjaxinline] or [mathjaxinline]x \rightarrow s^{-}[/mathjaxinline], where [mathjaxinline]s[/mathjaxinline] is a finite number that is a singularity of the function of interest. </p>
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<p><b class="bfseries">Worked example</b></p><p>
We know that [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x}[/mathjaxinline] is divergent. Use limit comparison to show that [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x^ p}[/mathjaxinline] is also divergent whenever [mathjaxinline]\displaystyle \frac{1}{x^ p}[/mathjaxinline] grows faster than [mathjaxinline]\displaystyle \frac{1}{x}[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]0[/mathjaxinline]. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Show worked solution<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p>
(As you read the solution, try to justify each step.) </p><p>
We can use that [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x}[/mathjaxinline] is divergent and limit comparison to see that [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x^ p}[/mathjaxinline] is also divergent whenever [mathjaxinline]\displaystyle \frac{1}{x^ p}[/mathjaxinline] grows faster than [mathjaxinline]\displaystyle \frac{1}{x}[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] approaches [mathjaxinline]0[/mathjaxinline]. </p><p>
When is [mathjaxinline]\displaystyle \frac{1}{x^ p} >> \frac{1}{x}[/mathjaxinline] as [mathjaxinline]x\rightarrow 0^{+}[/mathjaxinline]? </p><p>
To find out, we consider [mathjaxinline]\displaystyle \frac{1/x}{1/x^ p} = x^{p-1}[/mathjaxinline]. Note that </p><table id="a0000000935" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000936"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x^{p-1}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 0 \qquad \text {as } \quad x\rightarrow 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.100)</td></tr></table><p>
if [mathjaxinline]p-1>0[/mathjaxinline], which implies that [mathjaxinline]p>1[/mathjaxinline]. </p><p>
Thus by applying limit comparison, we see that knowing that since [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x}[/mathjaxinline] diverges, [mathjaxinline]\displaystyle \int _0^1 \frac{dx}{x^ p}[/mathjaxinline] must also diverge for [mathjaxinline]p\geq 1[/mathjaxinline]. </p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/9567be3f4e91361f26a8beaadf749715/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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Convergent and divergent improper integrals (*)
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Use comparison and limit comparison for improper integrals of type 1 and type 2 to determine which of the following improper integrals are <b class="bf">convergent</b>. </p>
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(Select all that apply. Note if some of your answers are correct, you will receive partial credit.) </p>
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<text>[mathjaxinline]\displaystyle \int _1^{\infty } \frac{dx}{\sqrt {x^3+5}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{0}^{1}\frac{dx}{x^3+x^2}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _1^{\infty } \frac{x^{2}dx}{x^3+2}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{0}^{1}\frac{dx}{\sqrt {1-x^3}}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{1}^{\infty }\frac{\ln (x) \, dx}{x^2}[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \int _{0}^{\infty }\frac{e^{-x}\, dx}{x}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">16. Infinite surface area and finite volume</h2>
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A shape with infinite surface area and finite volume
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Consider the shape obtained by revolving the curve [mathjaxinline]\ f(x) = 1/x[/mathjaxinline] for [mathjaxinline]1 \leq x &lt; \infty[/mathjaxinline] about the [mathjaxinline]x[/mathjaxinline]-axis. Find the surface area [mathjaxinline]A[/mathjaxinline] and volume [mathjaxinline]V[/mathjaxinline] of this shape. </p>
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(Enter numerical answers as math expressions or up to 2 decimal places. Enter <b class="bf">div</b> if the improper integral diverges.)<br/></p>
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<p style="display:inline">[mathjaxinline]A =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]V =[/mathjaxinline]</p>
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Consider the shape obtained by rotating the curve [mathjaxinline]\displaystyle \frac{1}{x^ p}[/mathjaxinline] for [mathjaxinline]1 \leq x \leq \infty[/mathjaxinline] about the [mathjaxinline]x[/mathjaxinline]-axis. Identify the value [mathjaxinline]M[/mathjaxinline] such that for [mathjaxinline]p\leq M[/mathjaxinline] the improper integral that defines the volume diverges. In this case, for [mathjaxinline]p=M[/mathjaxinline], does the surface area converge or diverge? </p>
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<p style="display:inline">[mathjaxinline]M =[/mathjaxinline]</p>
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<p style="display:inline">For [mathjaxinline]p=M[/mathjaxinline], &#8201;&#8201;&#8201; [mathjaxinline]A[/mathjaxinline] &#8195;</p>
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<h2 class="hd hd-2 unit-title">17. Summary</h2>
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<p><b class="bfseries">Review of l'Hopital's rule</b></p><p>
Here we recall l'Hôpital's rule in one case, [mathjaxinline]\displaystyle \frac{\infty }{\infty }[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \begin{cases} f(x) \longrightarrow \infty \\ g(x) \longrightarrow \infty \\ \frac{f'}{g'} \longrightarrow L \end{cases}[/mathjaxinline] as [mathjaxinline]x \rightarrow a[/mathjaxinline], </p><p>
then [mathjaxinline]\displaystyle \frac{f}{g} \longrightarrow L[/mathjaxinline] as [mathjaxinline]x \rightarrow a[/mathjaxinline]. </p><p>
(Recall that [mathjaxinline]a = \pm \infty[/mathjaxinline] and [mathjaxinline]L = \pm \infty[/mathjaxinline] is OK.) </p><p>
Consider [mathjaxinline]\ f, g >0[/mathjaxinline]. We say that [mathjaxinline]g[/mathjaxinline] grows faster than [mathjaxinline]\ f[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline], and write this as </p><table id="a0000000959" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle 0< f(x) << g(x),[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
if [mathjaxinline]\displaystyle \ f,g \longrightarrow \infty[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{f(x)}{g(x)} \longrightarrow 0[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><p><b class="bfseries">Examples</b></p><p>
For [mathjaxinline]p > 0,[/mathjaxinline] </p><p>
[mathjaxinline]\displaystyle \ln x << x^ p << e^ x << e^{x^2}.[/mathjaxinline] </p><p>
Note that [mathjaxinline]e^{x^2} = e^{(x^2)}[/mathjaxinline], not [mathjaxinline](e^ x)^2 = e^{2x}[/mathjaxinline]. </p><p><b class="bfseries">Rate of Decay (as [mathjaxinline]x→∞[/mathjaxinline])</b></p><p>
Consider [mathjaxinline]\ f, g >0[/mathjaxinline]. We say that [mathjaxinline]\ f[/mathjaxinline] decays faster than [mathjaxinline]g[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]\infty[/mathjaxinline], and write this as </p><table id="a0000000960" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle g(x) >> f(x)>0,[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
if [mathjaxinline]\displaystyle \ f,g \longrightarrow 0[/mathjaxinline] and [mathjaxinline]\displaystyle \frac{f(x)}{g(x)}\longrightarrow 0[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><p><b class="bfseries">Examples</b></p><p>
For [mathjaxinline]p >0,[/mathjaxinline] </p><table id="a0000000961" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{1}{\ln x} >> \frac{1}{x^ p} >> e^{-x} >> e^{-x^2}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Improper integrals definition</b></p><p>
An <span style="color:#27408C"><b class="bf">improper integral</b></span> is defined by [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx = \lim _{N\rightarrow \infty } \int _ a^{N} f(x) \, dx.[/mathjaxinline] </p><p>
This improper integral <span style="color:#27408C"><b class="bf">converges</b></span> if the limit exists and is finite.<br/>This improper integral <span style="color:#27408C"><b class="bf">diverges</b></span> if the limit does not exist (this includes when the limit is [mathjaxinline]\pm \infty[/mathjaxinline]). </p><p><b class="bfseries">Conclusion of example: powers of [mathjaxinline]x[/mathjaxinline]</b></p><p>
[mathjaxinline]\displaystyle \int _ a^{\infty } \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if } \, \, p\leq 1\\ \text {converges to} \, \, \frac{a^{-p+1}}{p-1} & \text {if }\, \, p> 1\end{cases}[/mathjaxinline] </p><p><b class="bfseries">Notation</b></p><p>
We say that [mathjaxinline]f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline] if [mathjaxinline]\displaystyle \frac{f(x)}{g(x)} \underset {x \rightarrow \infty }{\longrightarrow } 1[/mathjaxinline]. </p><p>
In words, we say that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] are <span style="color:#27408C"><b class="bf">similar</b></span> as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]. (The idea is that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] have the same asymptotic behavior as [mathjaxinline]x[/mathjaxinline] tends to infinity.) </p><p><b class="bfseries">Limit comparison</b></p><p>
The idea behind limit comparison is that if the asymptotic behavior is the same, then the improper integrals have the same behavior. </p><p>
If [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline], <br/>then the two integrals [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] and [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] (for large [mathjaxinline]a[/mathjaxinline]) either <span style="color:#27408C"><b class="bf">both converge</b></span> or <span style="color:#27408C"><b class="bf">both diverge</b></span>. </p><p>
This also works in the case where one function decays faster than the other as [mathjaxinline]x[/mathjaxinline] tends towards infinity. </p><p>
Suppose that [mathjaxinline]\ g(x)[/mathjaxinline] decays faster than [mathjaxinline]\ f(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow \infty[/mathjaxinline]. That is [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow \infty[/mathjaxinline]. </p><ul class="itemize"><li><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] converges. </p></li><li><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] diverges. </p></li></ul><p><b class="bfseries">Comparison</b></p><p>
Suppose [mathjaxinline]f(x) \geq g(x)>0[/mathjaxinline] for [mathjaxinline]x \geq a[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] converges also. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{\infty } g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{\infty } f(x) \, dx[/mathjaxinline] diverges also. </p><p><b class="bfseries">Definition of singularity</b></p><p>
A <span style="color:#27408C"><b class="bf">singularity</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] is a point [mathjaxinline]x=s[/mathjaxinline] such that the function [mathjaxinline]\ f(x)[/mathjaxinline] does not exist at [mathjaxinline]x=s.[/mathjaxinline] </p><p>
There are three main ways that the function can fail to exist at a point: </p><ul class="itemize"><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{+}} \left| f(x)\right|[/mathjaxinline] and/or [mathjaxinline]\displaystyle \lim _{x \rightarrow s^{-}} \left|f(x)\right|[/mathjaxinline] tends to [mathjaxinline]\infty[/mathjaxinline]. This is the case of most interest in this section. </p></li><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{\pm }} f(x)[/mathjaxinline] does not exist. In this case the function [mathjaxinline]\ f[/mathjaxinline] may oscillate, or have a jump discontinuity. </p></li><li><p>
[mathjaxinline]\displaystyle \lim _{x \rightarrow s^{\pm }} f(x)[/mathjaxinline] exists and is finite. In this case, the function [mathjaxinline]\ f[/mathjaxinline] has a removable discontinuity. </p></li></ul><p><b class="bfseries">Definition of improper integrals of the 2nd type</b></p><p>
An <span style="color:#27408C"><b class="bf">improper integral of the 2nd type</b></span> is an integral [mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx[/mathjaxinline] such that the function [mathjaxinline]\ f(x)[/mathjaxinline] has a singularity at [mathjaxinline]x=s[/mathjaxinline] for some [mathjaxinline]s[/mathjaxinline] with [mathjaxinline]a \leq s \leq b[/mathjaxinline]. </p><p>
For example, if [mathjaxinline]f(x)[/mathjaxinline] has a singularity at [mathjaxinline]x=b[/mathjaxinline], then </p><p>
[mathjaxinline]\displaystyle \int _ a^ b f(x) \, dx = \lim _{C \rightarrow b^{-}} \int _ a^ C f(x) \, dx.[/mathjaxinline] </p><p>
We say </p><ul class="itemize"><li><p>
the integral <span style="color:#27408C"><b class="bf">converges</b></span> if the limit exists and is finite. </p></li><li><p>
the integral <span style="color:#27408C"><b class="bf">diverges</b></span> if the limit does not exist (which includes the case that the limit is [mathjaxinline]\pm \infty[/mathjaxinline].) </p></li></ul><p><b class="bfseries">Overview of improper integrals</b></p><p>
The improper integral [mathjaxinline]\displaystyle \int _ a^{\infty } \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if}\, \, p\leq 1\\ \text {converges to} \, \, \frac{a^{1-p}}{p-1} & \text {if}\, \, p> 1\end{cases}[/mathjaxinline]. </p><p>
The improper integral [mathjaxinline]\displaystyle \int _0^{a} \frac{dx}{x^ p} \qquad \begin{cases} \text {diverges} & \text {if}\, \, p\geq 1\\ \text {converges to} \, \, \frac{a^{1-p}}{1-p} & \text {if}\, \, p< 1\end{cases}[/mathjaxinline] </p><div id="a0000000962" class="figure"><center><img src="/assets/courseware/v1/70ddb1e9817d840ec98831c4785d7b02/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_improper-01.svg" width="550px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 34</b>: <span> From left to right, we see the areas for [mathjaxinline]0 \leq x \leq 1[/mathjaxinline] and [mathjaxinline]1 \leq x < \infty[/mathjaxinline]<br/>under the graphs of [mathjaxinline]\displaystyle \frac{1}{x},\,[/mathjaxinline] [mathjaxinline]\displaystyle \frac{1}{x^2},\,[/mathjaxinline] and [mathjaxinline]\, \displaystyle \frac{1}{\sqrt {x}}[/mathjaxinline].<br/>The areas shaded in pink are infinite. The areas shaded in green are finite.</span></div></center></div><p><b class="bfseries">Comparison tests for improper integrals of 2nd type</b></p><p>
Suppose that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] both have a singularity at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
Suppose [mathjaxinline]f(x)>g(x)\geq 0[/mathjaxinline] for all [mathjaxinline]a\leq x \leq b[/mathjaxinline] except at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] converges also. </p><p>
If [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] diverges also. </p><p><b class="bfseries">Notation</b></p><p>
Suppose that [mathjaxinline]\ f(x)[/mathjaxinline] and [mathjaxinline]\ g(x)[/mathjaxinline] have a singularity at [mathjaxinline]x=s[/mathjaxinline]. ([mathjaxinline]\ f, \ g \longrightarrow \pm \infty[/mathjaxinline] as [mathjaxinline]x\rightarrow s^{+}[/mathjaxinline] and/or as [mathjaxinline]x\rightarrow s^{-}[/mathjaxinline].) </p><p>
We say that [mathjaxinline]\ f(x)[/mathjaxinline] is <span style="color:#27408C"><b class="bf">similar</b></span> to [mathjaxinline]\ g(x)[/mathjaxinline], and write [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow s^{+}[/mathjaxinline] or [mathjaxinline]x\rightarrow s^{-}[/mathjaxinline] if </p><table id="a0000000963" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000964"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \frac{f(x)}{g(x)}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 1 \qquad \text { as } \quad x \longrightarrow s^{\pm }.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(2.106)</td></tr></table><p>
We say that [mathjaxinline]\ f(x)[/mathjaxinline] grows faster than [mathjaxinline]\ g(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]s[/mathjaxinline], and write [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow s^{\pm }[/mathjaxinline], if [mathjaxinline]\displaystyle \ \ \begin{cases} f(x) \longrightarrow \infty \\ g(x) \longrightarrow \infty \\ \frac{g(x)}{f(x)} \longrightarrow 0 \end{cases}[/mathjaxinline] as [mathjaxinline]\displaystyle x \longrightarrow s^{\pm }.[/mathjaxinline] </p><p><b class="bfseries">Limit comparison tests for improper integrals of 2nd type</b></p><p>
Suppose that [mathjaxinline]f(x)[/mathjaxinline] and [mathjaxinline]g(x)[/mathjaxinline] both have a singularity at [mathjaxinline]x=s[/mathjaxinline]. </p><p>
Suppose [mathjaxinline]f(x), g(x)\geq 0[/mathjaxinline] for all [mathjaxinline]a\leq x \leq b[/mathjaxinline] except at [mathjaxinline]x=s[/mathjaxinline]. </p><ol class="enumerate"><li value="1"><p>
If [mathjaxinline]\ f(x) \sim g(x)[/mathjaxinline] as [mathjaxinline]x\rightarrow s^{\pm }[/mathjaxinline], <br/>then the two integrals [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] and [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] either <span style="color:#27408C"><b class="bf">both converge</b></span> or <span style="color:#27408C"><b class="bf">both diverge</b></span>. </p></li><li value="2"><p>
Suppose that [mathjaxinline]\ f(x)[/mathjaxinline] grows faster than [mathjaxinline]\ g(x)[/mathjaxinline] as [mathjaxinline]x[/mathjaxinline] tends towards [mathjaxinline]s^{\pm }[/mathjaxinline]. In other words, [mathjaxinline]\ f(x) >> g(x)[/mathjaxinline] as [mathjaxinline]x \rightarrow s^{\pm }[/mathjaxinline]. </p><ul class="itemize"><li><p>
If [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] converges, then [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] converges. </p></li><li><p>
If [mathjaxinline]\displaystyle \int _ a^{b} g(x) \, dx[/mathjaxinline] diverges, then [mathjaxinline]\displaystyle \int _ a^{b} f(x) \, dx[/mathjaxinline] diverges. </p></li></ul></li></ol><p>
Note that this notation is exactly the same notation that we had before. The only difference is that instead of having [mathjaxinline]x \rightarrow \infty[/mathjaxinline], we have [mathjaxinline]x \rightarrow s^{+}[/mathjaxinline] or [mathjaxinline]x \rightarrow s^{-}[/mathjaxinline], where [mathjaxinline]s[/mathjaxinline] is a finite number that is a singularity of the function of interest. </p>
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