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<div class="span10 offset1">
<h1>Improper Integrals</h1>
<p>A function <span class="math">$f(x)$</span> is Riemann integrable over an interval <span class="math">$[a,b]$</span> if some limit involving Riemann sums exists. This limit will fail to exist if <span class="math">$f(x) = \infty$</span> in <span class="math">$[a,b]$</span>. As well, the Riemann sum idea is undefined if either <span class="math">$a$</span> or <span class="math">$b$</span> (or both) are infinite, so the limit won't exist in this case.</p>
<p>To define integrals with either functions having singularities or infinite domains, the idea of an improper integral is introduced with definitions to handle the two cases above.</p>
<div class="well well-sm">
<figure>
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"/>
<figcaption><div class="markdown"><p>Area under $1/\sqrt{x}$ over $[a,b]$ increases as $a$ gets closer to $0$. Will it grow unbounded or have a limit?</p>
</div></figcaption>
</figure>
</div>
<h2>Infinite domains</h2>
<p>Let <span class="math">$f(x)$</span> be a reasonable function, so reasonable that for any <span class="math">$a < b$</span> the function is Riemann integrable, meaning <span class="math">$\int_a^b f(x)dx$</span> exists.</p>
<p>What needs to be the case so that we can discuss the limit over the entire real number line?</p>
<p>Clearly something. The function <span class="math">$f(x) = 1$</span> is reasonable by the idea above. Clearly the integral over and <span class="math">$[a,b]$</span> is just <span class="math">$b-a$</span>, but the limit over an unbounded domain would be <span class="math">$\infty$</span>. Even though limits of infinity can be of interest in some cases, not so here. What will ensure that the area is finite over an infinite region?</p>
<p>Or is that even the right question. Now consider <span class="math">$f(x) = \sin(\pi x)$</span>. Over every interval of the type <span class="math">$[-2n, 2n]$</span> the area is <span class="math">$0$</span>, and over any interval, <span class="math">$[a,b]$</span> the area never gets bigger than <span class="math">$2$</span>. But still this function does not have a well defined area on an infinite domain.</p>
<p>The right question involves a limit. Fix a finite <span class="math">$a$</span>. We define the definite integral over <span class="math">$[a,\infty)$</span> to be</p>
<p class="math">\[
~
\int_a^\infty f(x) dx = \lim_{M \rightarrow \infty} \int_a^M f(x) dx,
~
\]</p>
<p>when the limit exists. Similarly, we define the definite integral over <span class="math">$(-\infty, a]$</span> through</p>
<p class="math">\[
~
\int_{-\infty}^a f(x) dx = \lim_{M \rightarrow -\infty} \int_M^a f(x) dx.
~
\]</p>
<p>For the interval <span class="math">$(-\infty, \infty)$</span> we have need <em>both</em> these limits to exist, and then:</p>
<p class="math">\[
~
\int_{-\infty}^\infty f(x) dx = \lim_{M \rightarrow -\infty} \int_M^a f(x) dx + \lim_{M \rightarrow \infty} \int_a^M f(x) dx.
~
\]</p>
<div class="alert alert-info" role="alert">
<div class="markdown"><p>When the integral exists, it is said to <em>converge</em>. If it doesn't exist, it is said to <em>diverge</em>.</p>
</div>
</div>
<h5>Examples</h5>
<ul>
<li><p>The function <span class="math">$f(x) = 1/x^2$</span> is integrable over <span class="math">$[1, \infty)$</span>, as this limit exists:</p>
</li>
</ul>
<p class="math">\[
~
\lim_{M \rightarrow \infty} \int_1^M \frac{1}{x^2}dx = \lim_{M \rightarrow \infty} -\frac{1}{x}\big|_1^M
= \lim_{M \rightarrow \infty} 1 - \frac{1}{M} = 1.
~
\]</p>
<ul>
<li><p>The function <span class="math">$f(x) = 1/x^{1/2}$</span> is not integrable over <span class="math">$[1, \infty)$</span>, as this limit fails to exist:</p>
</li>
</ul>
<p class="math">\[
~
\lim_{M \rightarrow \infty} \int_1^M \frac{1}{x^{1/2}}dx = \lim_{M \rightarrow \infty} \frac{x^{1/2}}{1/2}\big|_1^M
= \lim_{M \rightarrow \infty} 2\sqrt{M} - 2 = \infty.
~
\]</p>
<p>The limit is infinite, so does not exist except in an extended sense.</p>
<ul>
<li><p>The function <span class="math">$x^n e^{-x}$</span> for <span class="math">$n = 1, 2, \dots$</span> is integrable over <span class="math">$[0,\infty)$</span>.</p>
</li>
</ul>
<p>Before showing this, we recall the fundamental theorem of calculus. The limit existing is the same as saying the limit of <span class="math">$F(M) - F(a)$</span> exists for an antiderivative of <span class="math">$f(x)$</span>.</p>
<p>For this particular problem, it can be shown by integration by parts that for positive, integer values of <span class="math">$n$</span> that an antiderivative exists of the form <span class="math">$F(x) = p(x)e^{-x}$</span>, where <span class="math">$p(x)$</span> is a polynomial of degree <span class="math">$n$</span>. But we've seen that for any <span class="math">$n>0$</span>, <span class="math">$\lim_{x \rightarrow \infty} x^n e^{-x} = 0$</span>, so the same is true for any polynomial. So, <span class="math">$\lim_{M \rightarrow \infty} F(M) - F(1) = -F(1)$</span>.</p>
<ul>
<li><p>The function <span class="math">$e^x$</span> is integrable over <span class="math">$(-\infty, a]$</span> but not</p>
</li>
</ul>
<p class="math">\[
[a, \infty)
\]</p>
<p>for any finite <span class="math">$a$</span>. This is because, <span class="math">$F(M) = e^x$</span> and this has a limit as <span class="math">$x$</span> goes to <span class="math">$-\infty$</span>, but not <span class="math">$\infty$</span>.</p>
<ul>
<li><p>Let <span class="math">$f(x) = x e^{-x^2}$</span>. This function has an integral over <span class="math">$[0, \infty)$</span> and more generally <span class="math">$(-\infty, \infty)$</span>. To see, we note that as it is an odd function, the area from <span class="math">$0$</span> to <span class="math">$M$</span> is the opposite sign of that from <span class="math">$-M$</span> to <span class="math">$0$</span>. So <span class="math">$\lim_{M \rightarrow \infty} (F(M) - F(0)) = \lim_{M \rightarrow -\infty} (F(0) - (-F(\lvert M\lvert)))$</span>. We only then need to investigate the one limit. But we can see by substitution with <span class="math">$u=x^2$</span>, that an antiderivative is <span class="math">$F(x) = (-1/2) \cdot e^{-x^2}$</span>. Clearly, <span class="math">$\lim_{M \rightarrow \infty}F(M) = 0$</span>, so the answer is well defined, and the area from <span class="math">$0$</span> to <span class="math">$\infty$</span> is just <span class="math">$e/2$</span>. From <span class="math">$-\infty$</span> to <span class="math">$0$</span> it is <span class="math">$-e/2$</span> and the total area is <span class="math">$0$</span>, as the two sides "cancel" out.</p>
</li>
<li><p>Let <span class="math">$f(x) = \sin(x)$</span>. Even though <span class="math">$\lim_{M \rightarrow \infty} (F(M) - F(-M) ) = 0$</span>, this function is not integrable. The fact is we need <em>both</em> the limit <span class="math">$F(M)$</span> and <span class="math">$F(-M)$</span> to exist as <span class="math">$M$</span> goes to <span class="math">$\infty$</span>. In this case, even though the area cancels if <span class="math">$\infty$</span> is approached at the same rate, this isn't sufficient to guarantee the two limits exists independently.</p>
</li>
</ul>
<ul>
<li><p>Will the function <span class="math">$f(x) = 1/(x\cdot(\log(x))^2)$</span> have an integral over <span class="math">$[e, \infty)$</span>?</p>
</li>
</ul>
<p>We first find an antiderivative using the <span class="math">$u$</span>-substitution <span class="math">$u(x) = \log(x)$</span>:</p>
<p class="math">\[
~
\int_e^M \frac{e}{x \log(x)^{2}} dx
= \int_{\log(e)}^{\log(M)} \frac{1}{u^{2}} du
= \frac{-1}{u} \big|_{1}^{\log(M)}
= \frac{-1}{\log(M)} - \frac{-1}{1}
= 1 - \frac{1}{M}.
~
\]</p>
<p>As <span class="math">$M$</span> goes to <span class="math">$\infty$</span>, this will converge to <span class="math">$1$</span>.</p>
<ul>
<li><p>The sinc function <span class="math">$f(x) = \sin(\pi x)/(\pi x)$</span> does not have a nice antiderivative. Seeing if the limit exists is a bit of a problem. However, this function is important enough that there is a built-in function, <code>Si</code>, that computes <span class="math">$\int_0^x \sin(u)/u\cdot du$</span>. This function can be used through <code>sympy.Si(...)</code>:</p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CalculusWithJulia</span><span class='hljl-t'> </span><span class='hljl-cs'># loads `SymPy`, `QuadGK`</span><span class='hljl-t'>
</span><span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>M</span><span class='hljl-t'>
</span><span class='hljl-nf'>limit</span><span class='hljl-p'>(</span><span class='hljl-n'>sympy</span><span class='hljl-oB'>.</span><span class='hljl-nf'>Si</span><span class='hljl-p'>(</span><span class='hljl-n'>M</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-n'>M</span><span class='hljl-t'> </span><span class='hljl-oB'>=></span><span class='hljl-t'> </span><span class='hljl-n'>oo</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}\frac{\pi}{2}\end{equation*}</div>
<h3>Numeric integrals</h3>
<p>The <code>quadgk</code> function (available through <code>QuadGK</code> which is loaded with <code>CalculusWithJulia</code>) is able to accept <code>Inf</code> and <code>-Inf</code> as endpoints of the interval. For example, this will integrate <span class="math">$e^{-x^2/2}$</span> over the real line:</p>
<pre class='hljl'>
<span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>exp</span><span class='hljl-p'>(</span><span class='hljl-oB'>-</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-oB'>/</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>quadgk</span><span class='hljl-p'>(</span><span class='hljl-n'>f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-n'>Inf</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Inf</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
(2.506628274639168, 3.6084380708526996e-8)
</pre>
<p>(If may not be obvious, but this is <span class="math">$\sqrt{2\pi}$</span>.)</p>
<h2>Singularities</h2>
<p>Suppose <span class="math">$\lim_{x \rightarrow c}f(x) = \infty$</span> or <span class="math">$-\infty$</span>. Then a Riemann sum that contains an interval including <span class="math">$c$</span> will not be finite if the point chosen in the interval is <span class="math">$c$</span>. Though we could choose another point, this is not enough as the definition must hold for any choice of the <span class="math">$c_i$</span>.</p>
<p>However, if <span class="math">$c$</span> is isolated, we can get close to <span class="math">$c$</span> and see how the area changes.</p>
<p>Suppose <span class="math">$a < c$</span>, we define <span class="math">$\int_a^c f(x) dx = \lim_{M \rightarrow c-} \int_a^c f(x) dx$</span>. If this limit exists, the definite integral with <span class="math">$c$</span> is well defined. Similarly, the integral from <span class="math">$c$</span> to <span class="math">$b$</span>, where <span class="math">$b > c$</span>, can be defined by a right limit going to <span class="math">$c$</span>. The integral from <span class="math">$a$</span> to <span class="math">$b$</span> will exist if both the limits are finite.</p>
<h5>Examples</h5>
<ul>
<li><p>Consider the example of the initial illustration, <span class="math">$f(x) = 1/\sqrt{x}$</span> at <span class="math">$0$</span>. Here <span class="math">$f(0)= \infty$</span>, so the usual notion of a limit won't apply to <span class="math">$\int_0^1 f(x) dx$</span>. However,</p>
</li>
</ul>
<p class="math">\[
~
\lim_{M \rightarrow 0+} \int_M^1 \frac{1}{\sqrt{x}} dx
= \lim_{M \rightarrow 0+} \frac{\sqrt{x}}{1/2} \big|_M^1
= \lim_{M \rightarrow 0+} 2(1) - 2\sqrt{M} = 2.
~
\]</p>
<div class="alert alert-info" role="alert">
<div class="markdown"><p>The cases $f(x) = x^{-n}$ for $n > 0$ are tricky. For $n > 1$, the functions can be integrated over $[1,\infty)$, but not $(0,1]$. For $0 < n < 1$, the functions can be integrated over $(0,1]$ but not $[1, \infty)$.</p>
</div>
</div>
<ul>
<li><p>Now consider <span class="math">$f(x) = 1/x$</span>. Is this integral <span class="math">$\int_0^1 1/x \cdot dx$</span> defined? It will be <em>if</em> this limit exists:</p>
</li>
</ul>
<p class="math">\[
~
\lim_{M \rightarrow 0+} \int_M^1 \frac{1}{x} dx
= \lim_{M \rightarrow 0+} \log(x) \big|_M^1
= \lim_{M \rightarrow 0+} \log(1) - \log(M) = \infty.
~
\]</p>
<p>As the limit does not exist, the function is not integrable around <span class="math">$0$</span>.</p>
<ul>
<li><p><code>SymPy</code> may give answers which do not coincide with our definitions, as it uses complex numbers as a default assumption. In this case it returns <code>NaN</code> for an integral with no answer:</p>
</li>
</ul>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'>
</span><span class='hljl-nf'>integrate</span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>/</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-p'>))</span>
</pre>
<div class="well well-sm">\begin{equation*}\mathrm{NaN}\end{equation*}</div>
<ul>
<li><p>Suppose you know <span class="math">$\int_1^\infty x^2 f(x) dx$</span> exists. Does this imply <span class="math">$\int_0^1 f(1/x) dx$</span> exists?</p>
</li>
</ul>
<p>We need to consider the limit of <span class="math">$\int_M^1 f(1/x) dx$</span>. We try the <span class="math">$u$</span>-substitution <span class="math">$u(x) = 1/x$</span>. This gives <span class="math">$du = -(1/x^2)dx = -u^2 dx$</span>. So, the substitution becomes:</p>
<p class="math">\[
~
\int_M^1 f(1/x) dx = \int_{1/M}^{1/1} f(u) (-u^2) du = \int_1^{1/M} u^2 f(u) du.
~
\]</p>
<p>But the limit as <span class="math">$M \rightarrow 0$</span> of <span class="math">$1/M$</span> is the same going to <span class="math">$\infty$</span>, so the right side will converge by the assumption. Thus we get <span class="math">$f(1/x)$</span> is integrable over <span class="math">$(0,1]$</span>.</p>
<h3>Numeric integration</h3>
<p>So far our use of the <code>quadgk</code> function specified the region to integrate via <code>a</code>, <code>b</code>, as in <code>quadgk(f, a, b)</code>. In fact, it can specify values in between for which the function should not be sampled. For example, were we to integrate <span class="math">$1/\sqrt{\lvert x\rvert}$</span> over <span class="math">$[-1,1]$</span>, we would want to avoid <span class="math">$0$</span> as a point to sample. Here is how:</p>
<pre class='hljl'>
<span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-t'> </span><span class='hljl-oB'>/</span><span class='hljl-t'> </span><span class='hljl-nf'>sqrt</span><span class='hljl-p'>(</span><span class='hljl-nf'>abs</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-nf'>quadgk</span><span class='hljl-p'>(</span><span class='hljl-n'>f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-p'>)</span>
</pre>
<pre class="output">
(3.999999962817228, 5.736423067171012e-8)
</pre>
<p>Just trying <code>quadgk(f, -1, 1)</code> leads to a <code>DomainError</code>, as <code>0</code> will be one of the points sampled. The general call is like <code>quadgk(f, a, b, c, d,...)</code> which integrates over <span class="math">$(a,b)$</span> and <span class="math">$(b,c)$</span> and <span class="math">$(c,d)$</span>, <span class="math">$\dots$</span>. The algorithm is not supposed to evaluate the function at the endpoints of the intervals.</p>
<h2>Probability applications</h2>
<p>A probability density is a function <span class="math">$f(x) \geq 0$</span> which is integrable on <span class="math">$(-\infty, \infty)$</span> and for which <span class="math">$\int_{-\infty}^\infty f(x) dx =1$</span>. The cumulative distribution function is defined by <span class="math">$F(x)=\int_{-\infty}^x f(u) du$</span>.</p>
<p>Probability densities are good example of using improper integrals.</p>
<ul>
<li><p>Show that <span class="math">$f(x) = (1/\pi) 1/(1 + x^2)$</span> is a probability density function.</p>
</li>
</ul>
<p>We need to show that the integral exists and is <span class="math">$1$</span>. For this, we use the fact that <span class="math">$(1/\pi) \cdot \tan^{-1}(x)$</span> is an antiderivative. Then we have:</p>
<p class="math">\[
\lim_{M \rightarrow \infty} F(M) = (1/\pi) \cdot \pi/2
\]</p>
<p>and as <span class="math">$\tan^{-1}(x)$</span> is odd, we must have <span class="math">$F(-\infty) = \lim_{M \rightarrow -\infty} f(M) = -(1/\pi) \cdot \pi/2$</span>. All told, <span class="math">$F(\infty) - F(-\infty) = 1/2 - (-1/2) = 1$</span>.</p>
<ul>
<li><p>Show that <span class="math">$f(x) = 1/(b-a)$</span> for <span class="math">$a \leq x \leq b$</span> and <span class="math">$0$</span> otherwise is a probability density.</p>
</li>
</ul>
<p>The integral for <span class="math">$-\infty$</span> to <span class="math">$a$</span> of <span class="math">$f(x)$</span> is just an integral of the constant <span class="math">$0$</span>, so will be <span class="math">$0$</span>. (This is the only constant with finite area over an infinite domain.) Similarly, the integral from <span class="math">$b$</span> to <span class="math">$\infty$</span> will be <span class="math">$0$</span>. This means:</p>
<p class="math">\[
~
\int_{-\infty}^\infty f(x) dx = \int_a^b \frac{1}{b-a} dx = 1.
~
\]</p>
<ul>
<li><p>Show that if <span class="math">$f(x)$</span> is a probability density then so is <span class="math">$f(x-c)$</span> for any <span class="math">$c$</span>.</p>
</li>
</ul>
<p>We have by the <span class="math">$u$</span>-substitution</p>
<p class="math">\[
~
\int_{-\infty}^\infty f(x-c)dx = \int_{u(-\infty)}^{u(\infty)} f(u) du = \int_{-\infty}^\infty f(u) du = 1.
~
\]</p>
<p>The key is that we can use the regular <span class="math">$u$</span>-substitution formula provided <span class="math">$\lim_{M \rightarrow \infty} u(M) = u(\infty)$</span> is defined. (The <em>informal</em> notation <span class="math">$u(\infty)$</span> is defined by that limit.)</p>
<ul>
<li><p>If <span class="math">$f(x)$</span> is a probability density, then so is <span class="math">$(1/h) f((x-c)/h)$</span> for any <span class="math">$c, h > 0$</span>.</p>
</li>
</ul>
<p>Again, by a <span class="math">$u$</span> substitution with, now, <span class="math">$u(x) = (x-c)/h$</span>, we have <span class="math">$du = (1/h) \cdot dx$</span> and the result follows just as before:</p>
<p class="math">\[
~
\int_{-\infty}^\infty \frac{1}{h}f(\frac{x-c}{h})dx = \int_{u(-\infty)}^{u(\infty)} f(u) du = \int_{-\infty}^\infty f(u) du = 1.
~
\]</p>
<ul>
<li><p>If <span class="math">$F(x) = 1 - e^{-x}$</span>, for <span class="math">$x \geq 0$</span>, and <span class="math">$0$</span> otherwise, find <span class="math">$f(x)$</span>.</p>
</li>
</ul>
<p>We want to just say <span class="math">$F'(x)= e^{-x}$</span> so <span class="math">$f(x) = e^{-x}$</span>. But some care is needed. First, that isn't right. The derivative for <span class="math">$x<0$</span> of <span class="math">$F(x)$</span> is <span class="math">$0$</span>, so <span class="math">$f(x) = 0$</span> if <span class="math">$x < 0$</span>. What about for <span class="math">$x>0$</span>? The derivative is <span class="math">$e^{-x}$</span>, but is that the right answer? <span class="math">$F(x) = \int_{-\infty}^x f(u) du$</span>, so we have to at least discuss if the <span class="math">$-\infty$</span> affects things. In this case, and in general the answer is <em>no</em>. For any <span class="math">$x$</span> we can find <span class="math">$M < x$</span> so that we have <span class="math">$F(x) = \int_{-\infty}^M f(u) du + \int_M^x f(u) du$</span>. The first part is a constant, so will have derivative <span class="math">$0$</span>, the second will have derivative <span class="math">$f(x)$</span>, if the derivative exists (and it will exist at <span class="math">$x$</span> if the derivative is continuous in a neighborhood of <span class="math">$x$</span>).</p>
<p>Finally, at <span class="math">$x=0$</span> we have an issue, as <span class="math">$F'(0)$</span> does not exist. The left limit of the secant line approximation is <span class="math">$0$</span>, the right limit of the secant line approximation is <span class="math">$1$</span>. So, we can take <span class="math">$f(x) = e^{-x}$</span> for <span class="math">$x > 0$</span> and <span class="math">$0$</span> otherwise, noting that redefining <span class="math">$f(x)$</span> at a point will not effect the integral as long as the point is finite.</p>
<h2>Questions</h2>
<h6>Question</h6>
<p>Is <span class="math">$f(x) = 1/x^{100}$</span> integrable around <span class="math">$0$</span>?</p>
<form name="WeaveQuestion" data-id="i1UWu44N" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_i1UWu44N" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_i1UWu44N" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="i1UWu44N_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_i1UWu44N']").on("change", function() {
correct = this.value == 2;
if(correct) {
$("#i1UWu44N_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#i1UWu44N_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6>
<p>Is <span class="math">$f(x) = 1/x^{1/3}$</span> integrable around <span class="math">$0$</span>?</p>
<form name="WeaveQuestion" data-id="4tswXts8" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_4tswXts8" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_4tswXts8" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="4tswXts8_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_4tswXts8']").on("change", function() {
correct = this.value == 1;
if(correct) {
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} else {
$("#4tswXts8_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6>
<p>Is <span class="math">$f(x) = x\cdot\log(x)$</span> integrable on <span class="math">$[1,\infty)$</span>?</p>
<form name="WeaveQuestion" data-id="t68Ex94r" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_t68Ex94r" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_t68Ex94r" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="t68Ex94r_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_t68Ex94r']").on("change", function() {
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} else {
$("#t68Ex94r_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6>
<p>Is <span class="math">$f(x) = \log(x)/ x$</span> integrable on <span class="math">$[1,\infty)$</span>?</p>
<form name="WeaveQuestion" data-id="jnlaEyAJ" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_jnlaEyAJ" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_jnlaEyAJ" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="jnlaEyAJ_message"></div>
</div>
</form>
<script text="text/javascript">
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} else {
$("#jnlaEyAJ_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6>
<p>Is <span class="math">$f(x) = \log(x)$</span> integrable on <span class="math">$[1,\infty)$</span>?</p>
<form name="WeaveQuestion" data-id="Mc05RCqM" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_Mc05RCqM" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_Mc05RCqM" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="Mc05RCqM_message"></div>
</div>
</form>
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$("#Mc05RCqM_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
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});
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<h6>Question</h6>
<p>Compute the integral <span class="math">$\int_0^\infty 1/(1+x^2) dx$</span>.</p>
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</div>
</div>
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<h6>Question</h6>
<p>Compute the the integral <span class="math">$\int_1^\infty \log(x)/x^2 dx$</span>.</p>
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<h6>Question</h6>
<p>Compute the integral <span class="math">$\int_0^2 (x-1)^{2/3} dx$</span>.</p>
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}
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</script>
<h6>Question</h6>
<p>From the relationship that if <span class="math">$0 \leq f(x) \leq g(x)$</span> then <span class="math">$\int_a^b f(x) dx \leq \int_a^b g(x) dx$</span> it can be deduced that</p>
<ul>
<li><p>if <span class="math">$\int_a^\infty f(x) dx$</span> diverges, then so does <span class="math">$\int_a^\infty g(x) dx$</span>.</p>
</li>
<li><p>if <span class="math">$\int_a^\infty g(x) dx$</span> converges, then so does <span class="math">$\int_a^\infty f(x) dx$</span>.</p>
</li>
</ul>
<p>Let <span class="math">$f(x) = \lvert \sin(x)/x^2 \rvert$</span>.</p>
<p>What can you say about <span class="math">$\int_1^\infty f(x) dx$</span>, as <span class="math">$f(x) \leq 1/x^2$</span> on <span class="math">$[1, \infty)$</span>?</p>
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<label>
<input type="radio" name="radio_7VIsfWu8" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_7VIsfWu8" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_7VIsfWu8" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="7VIsfWu8_message"></div>
</div>
</form>
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}
});
</script>
<hr />
<p>Let <span class="math">$f(x) = \lvert \sin(x) \rvert / x$</span>.</p>
<p>What can you say about <span class="math">$\int_1^\infty f(x) dx$</span>, as <span class="math">$f(x) \leq 1/x$</span> on <span class="math">$[1, \infty)$</span>?</p>
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<div class="form-group ">
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<label>
<input type="radio" name="radio_LUUevswN" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_LUUevswN" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_LUUevswN" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="LUUevswN_message"></div>
</div>
</form>
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} else {
$("#LUUevswN_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<hr />
<p>Let <span class="math">$f(x) = 1/\sqrt{x^2 - 1}$</span>. What can you say about <span class="math">$\int_1^\infty f(x) dx$</span>, as <span class="math">$f(x) \geq 1/x$</span> on <span class="math">$[1, \infty)$</span>?</p>
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<div class="form-group ">
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<label>
<input type="radio" name="radio_wGeTsWLt" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_wGeTsWLt" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_wGeTsWLt" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="wGeTsWLt_message"></div>
</div>
</form>
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}
});
</script>
<hr />
<p>Let <span class="math">$f(x) = 1 + 4x^2$</span>. What can you say about <span class="math">$\int_1^\infty f(x) dx$</span>, as <span class="math">$f(x) \leq 1/x^2$</span> on <span class="math">$[1, \infty)$</span>?</p>
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<div class="form-group ">
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<label>
<input type="radio" name="radio_8rosJ6Nu" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_8rosJ6Nu" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_8rosJ6Nu" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="8rosJ6Nu_message"></div>
</div>
</form>
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$("#8rosJ6Nu_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<hr />
<p>Let <span class="math">$f(x) = \lvert \sin(x)^{10}\rvert/e^x$</span>. What can you say about <span class="math">$\int_1^\infty f(x) dx$</span>, as <span class="math">$f(x) \leq e^{-x}$</span> on <span class="math">$[1, \infty)$</span>?</p>
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<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_0Q1uHNZM" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_0Q1uHNZM" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_0Q1uHNZM" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="0Q1uHNZM_message"></div>
</div>
</form>
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}
});
</script>
<h6>Question</h6>
<p>The difference between "blowing up" at <span class="math">$0$</span> versus being integrable at <span class="math">$\infty$</span> can be seen to be related through the <span class="math">$u$</span>-substitution <span class="math">$u=1/x$</span>. With this <span class="math">$u$</span>-substitution, what becomes of <span class="math">$\int_0^1 x^{-2/3} dx$</span>?</p>
<form name="WeaveQuestion" data-id="fhfJweeI" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_fhfJweeI" value="1"><div class="markdown">$\int_0^\infty 1/u \cdot du$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_fhfJweeI" value="2"><div class="markdown">$\int_0^1 u^{2/3} \cdot du$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_fhfJweeI" value="3"><div class="markdown">$\int_1^\infty u^{2/3}/u^2 \cdot du$
</div>
</label>
</div>
<div id="fhfJweeI_message"></div>
</div>
</form>
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} else {
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}
});
</script>
<h6>Question</h6>
<p>The antiderivative of <span class="math">$f(x) = 1/\pi \cdot 1/\sqrt{x(1-x)}$</span> is <span class="math">$F(x)=(2/\pi)\cdot \sin^{-1}(\sqrt{x})$</span>.</p>
<p>Find <span class="math">$\int_0^1 f(x) dx$</span>.</p>
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}
});
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