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<h1>Improper Integrals</h1><p>A function $f(x)$ is Riemann integrable over an interval $[a,b]$ if some limit involving Riemann sums is defined. This limit will fail if $f(x) = \infty$ in $[a,b]$. As well, the Riemann sum idea is undefined if either $a$ or $b$ (or both) are infinite.</p><p>To work with such cases, the idea of an improper integral is introduced with definitions to handle the two cases above: singularities in the function and infinite domains.</p><p><div class="well well-sm"> <figure> <img src="data:image/gif;base64, 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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 grown unbounded or have a limit?</p> </div></figcaption> </figure> </div></p><h2>Infinite domains</h2><p>Let $f(x)$ be a reasonable function, so reasonable that for any $a < b$ the function is Riemann integrable so that $\int_a^b f(x)dx$ 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 $f(x) = 1$ is reasonable by the idea above. Clearly the integral over and $[a,b]$ is just $b-a$, but the limit over an unbounded domain would be $\infty$. 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 $f(x) = \sin(\pi x)$. Over every interval of the type $[-2n, 2n]$ the area is $0$, and over any interval, $[a,b]$ the area never gets bigger than $2$. 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 $a$. We define the definite integral over $[a,\infty)$ to be</p>$$~
\int_a^\infty f(x) dx = \lim_{M \rightarrow \infty} \int_a^M f(x) dx,
~$$<p>when the limit exists. Similarly, we define the definite integral over $(-\infty, a]$ through</p>$$~
\int_{-\infty}^a f(x) dx = \lim_{M \rightarrow -\infty} \int_M^a f(x) dx.
~$$<p>For the interval $(-\infty, \infty)$ we have need both these limits to exist:</p>$$~
\int_{-\infty}^\infty f(x) dx = \lim_{M \rightarrow -\infty} \int_M^a f(x) dx + \lim_{M \rightarrow \infty} \int_a^M f(x) dx.
~$$<h5>Examples</h5><ul>
<li>The function $f(x) = 1/x^2$ is integrable over $[1, \infty)$, as this limit exists:</li>
</ul>$$~
\lim_{M \rightarrow \infty} \int_1^M \frac{1}{x^2}dx = \lim_{M \rightarrow \infty} -\frac{1}{x}\big|_1^M
= \lim_{M \rightarrow \infty} 1 - \frac{1}{M} = 1.
~$$<ul>
<li>The function $f(x) = 1/x^{1/2}$ is not integrable over $[1, \infty)$, as this limit fails to exist:</li>
</ul>$$~
\lim_{M \rightarrow \infty} \int_1^M \frac{1}{x^{1/2}}dx = \lim_{M \rightarrow \infty} \frac{x^{1/2}}{1/2}\big|_1^M
= \lim_{M \rightarrow \infty} 2\sqrt{M} - 2 = \infty.
~$$<p>The limit is infinite, so does not exist except in an extended sense.</p><ul>
<li>The function $x^n e^{-x}$ for $n = 1, 2, \dots$ is integrable over $[0,\infty)$.</li>
</ul><p>Before showing this, we recall the fundamental theorem of calculus. The limit existing is the same as saying the limit of $F(M) - F(a)$ exists for an antiderivative of $f(x)$.</p><p>For this particular problem, it can be shown by integration by parts that for positive, integer values of $n$ that an antiderivative exists of the form $F(x) = p(x)e^{-x}$, where $p(x)$ is a polynomial of degree $n$. But we've seen that for any $n>0$, $\lim_{x \rightarrow \infty} x^n e^{-x} = 0$, so the same is true for any polynomial. So, $\lim_{M \rightarrow \infty} F(M) - F(1) = -F(1)$.</p><ul>
<li>The function $e^x$ is integrable over $(-\infty, a]$ but not $[a, \infty)$ for any finite $a$. This is because, $F(M) = e^x$ and this has a limit as $x$ goes to $-\infty$, but not $\infty$.</li>
</ul><ul>
<li>Let $f(x) = x e^{-x^2}$. This function has an integral over $[0, \infty)$ and more generally $(-\infty, \infty)$. To see, we note that as it is an odd function, the area from $0$ to $M$ is the opposite sign of that from $-M$ to $0$. So $\lim_{M \rightarrow \infty} (F(M) - F(0)) = \lim_{M \rightarrow -\infty} (F(0) - (-F(\lvert M\lvert)))$. We only then need to investigate the one limit. But we can see by substitution with $u=x^2$, that an antiderivative is $F(x) = (-1/2) \cdot e^{-x^2}$. Clearly, $\lim_{M \rightarrow \infty}F(M) = 0$, so the answer is well defined, and the area from $0$ to $\infty$ is just $e/2$. From $-\infty$ to $0$ it is $-e/2$ and the total area is $0$, as the two sides "cancel" out.</li>
</ul><ul>
<li>Let $f(x) = \sin(x)$. Even though $\lim_{M \rightarrow \infty} (F(M) - F(-M) ) = 0$, this function is not integrable. The fact is we need both the limit $F(M)$ and $F(-M)$ to exist as $M$ goes to $\infty$. In this case, even though the area cancels if $\infty$ is approached at the same rate, this isn't sufficient to guarantee the two limits exists independently.</li>
</ul><ul>
<li>Will the function $f(x) = 1/(x\cdot(\log(x))^2)$ have an integral over $[e, \infty)$?</li>
</ul><p>We first find an antiderivative using the $u$-substitution $u(x) = \log(x)$:</p>$$~
\int_e^M \frac{e}{x \log(x)^{2}} dx
= \int_{\log(e)}^{\log(M)} \frac{1}{u^{2}} du
= \frac{-1}{u} \big|_{1}^{\log(M)}
= \frac{-1}{\log(M)} - \frac{-1}{1}
= 1 - \frac{1}{M}.
~$$<p>As $M$ goes to $\infty$, this will converge to $1$.</p><ul>
<li>The sinc function $f(x) = \sin(\pi x)/(\pi x)$ does not have a nice antiderivative. so 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>sinc</code>, that computes $\int_0^x f(u) du$. We can use this built in to investigate the limit:</li>
</ul><pre class="sourceCode julia">using SymPy
x, M = symbols("x, M", real=true)
integrate(sinc(x), (x, 0, M))</pre>
<div class="well well-sm">
$$\frac{1}{\pi} \operatorname{Si}{\left (\pi M \right )}$$</div>
<pre class="sourceCode julia">limit(integrate(sinc(x), (x, 0, M)), M, oo)</pre>
<div class="well well-sm">
$$\frac{1}{2}$$</div>
<h3>Numeric integrals</h3><p>The <code>quadgk</code> function is able to accept <code>Inf</code> and <code>-Inf</code> as endpoints of the interval. For example, this will integrate $e^{-x^2/2}$ over the real line:</p><pre class="sourceCode julia">f(x) = exp(-x^2/2)
quadgk(f, -Inf, Inf)</pre>
<pre class="output">
(2.506628274639168,3.6084380708526996e-8)</pre>
<p>(If may not be obvious, but this is $\sqrt{2\pi}$.)</p><h2>Singularities</h2><p>Suppose $\lim_{x \rightarrow c}f(x) = \infty$ or $-\infty$. Then a Riemann sum that contains an interval including $c$ will not be finite if the point chosen in the interval is $c$. Though we could choose another point, this is not enough as the definition must hold for any choice of the $c_i$.</p><p>However, if $c$ is isolated, we can get close to $c$ and see how the area changes.</p><p>Suppose $a < c$, we define $\int_a^c f(x) dx = \lim_{M \rightarrow c-} \int_a^c f(x) dx$. If this limit exists, the definite integral with $c$ is well defined. Similarly, the integral from $c$ to $b$, where $b > c$, can be defined by a right limit going to $c$. The integral from $a$ to $b$ will exist if both the limits are finite.</p><h5>Examples</h5><ul>
<li>Consider $f(x) = 1/\sqrt{x}$ at $0$. Here $f(0)= \infty$, so the usual notion of a limit won't apply to $\int_0^1 f(x) dx$. However,</li>
</ul>$$~
\lim_{M \rightarrow 0+} \int_M^1 \frac{1}{\sqrt{x}} dx
= \lim_{M \rightarrow 0+} \frac{\sqrt{x}}{1/2} \big|_M^1
= \lim_{M \rightarrow 0+} 2(1) - 2\sqrt{M} = 2.
~$$<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>Now consider $f(x) = 1/x$. Is this integral defined $\int_0^1 1/x dx$? It will be <em>if</em> this limit exists:</li>
</ul>$$~
\lim_{M \rightarrow 0+} \int_M^1 \frac{1}{x} dx
= \lim_{M \rightarrow 0+} \log(x) \big|_M^1
= \lim_{M \rightarrow 0+} \log(1) - \log(M) = \infty.
~$$<p>So the function is not integrable around $0$.</p><ul>
<li><code>SymPy</code> may give answers which do not coincide with our definitions, as it uses complex numbers as a default assumption. This is why this integral produces an imaginary number for an answer, even though we just saw that it is undefined:</li>
</ul><pre class="sourceCode julia">integrate(1/x, (x, -1, 1))</pre>
<div class="well well-sm">
$$- i \pi$$</div>
<ul>
<li>Suppose you know $\int_1^\infty x^2 f(x) dx$ exists. Does this imply $\int_0^1 f(1/x) dx$ exists?</li>
</ul><p>We need to consider the limit of $\int_M^1 f(1/x) dx$. We try the $u$-substitution $u(x) = 1/x$. This gives $du = -(1/x^2)dx = -u^2 dx$. So, the substitution becomes:</p>$$~
\int_M^1 f(1/x) dx = \int_{1/M}^{1/1} f(u) (-u^2) du = \int_1^{1/M} u^2 f(u) du.
~$$<p>But the limit as $M \rightarrow 0$ of $1/M$ is the same going to $\infty$, so the right side will converge by the assumption. Thus we get $f(1/x)$ is integrable over $(0,1]$.</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 $1/\sqrt{\lvert x\rvert}$ over $[-1,1]$, we would want to avoid $0$ as a point to sample. Here is how:</p><pre class="sourceCode julia">f(x) = 1 / sqrt(abs(x))
quadgk(f, -1, 0, 1)</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 $(a,b)$ and $(b,c)$ and $(c,d)$, $\dots$. 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 $f(x) \geq 0$ which is integrable on $(-\infty, \infty)$ and for which $\int_{-\infty}^\infty f(x) dx = 0$. The cumulative distribution function is defined by $F(x) = -\int_{-\infty}^x f(u) du$.</p><p>Probability densities are good example of using improper integrals.</p><ul>
<li>Show that $f(x) = (1/\pi) 1/(1 + x^2)$ is a probability density function.</li>
</ul><p>We need to show that the integral exists and is $1$. For this, we use the fact that $1/\pi \tan^{-1}(x)$ is an antiderivative. Then we have:</p>$$\lim_{M \rightarrow \infty} F(M) = (1/\pi) \cdot \pi/2$$<p>and as $\tan^{-1}(x)$ is odd, we must have $F(-\infty) = \lim_{M \rightarrow -\infty} f(M) = -(1/\pi) \cdot \pi/2$. All told, $F(\infty) - F(-\infty) = 1/2 - (-1/2) = 1$.</p><ul>
<li>Show that $f(x) = 1/(b-a)$ for $a \leq x \leq b$ and $0$ otherwise is a probability density.</li>
</ul><p>The integral for $-\infty$ to $a$ of $f(x)$ is just an integral of the constant $0$, so will be $0$. (This is the only constant with finite area over an infinite domain.) Similarly, the integral from $b$ to $\infty$ will be $0$. This means:</p>$$~
\int_{-\infty}^\infty f(x) dx = \int_a^b \frac{1}{b-a} dx = 1.
~$$<ul>
<li>Show that if $f(x)$ is a probability density then so is $f(x-c)$ for any $c$.</li>
</ul><p>We have by the $u$-substitution</p>$$~
\int_{-\infty}^\infty f(x-c)dx = \int_{u(-\infty)}^{u(\infty)} f(u) du = \int_{-\infty}^\infty f(u) du = 1.
~$$<p>The key is that we can use the regular $u$-substitution formula provided $\lim_{M \rightarrow \infty} u(M) = u(\infty)$ is defined. (The <em>informal</em> notation $u(\infty)$ is defined by that limit.)</p><ul>
<li>If $f(x)$ is a probability density, then so is $(1/h) f((x-c)/h)$ for any $c, h > 0$.</li>
</ul><p>Again, by a $u$ substitution with, now, $u(x) = (x-c)/h$, we have $du = 1/h dx$ and the result follows just as before:</p>$$~
\int_{-\infty}^\infty \frac{1}{h}f(\frac{x-c}{h})dx = \int_{u(-\infty)}^{u(\infty)} f(u) du = \int_{-\infty}^\infty f(u) du = 1.
~$$<ul>
<li>If $F(x) = 1 - e^{-x}$, for $x \geq 0$, and $0$ otherwise, find $f(x)$.</li>
</ul><p>We want to just say $F'(x)= e^{-x}$ so $f(x) = e^{-x}$. But some care is needed. First, that isn't right. The derivative for $x<0$ of $F(x)$ is $0$, so $f(x) = 0$ if $x < 0$. What about for $x>0$? The derivative is $e^{-x}$, but is that the right answer? $F(x) = \int_{-\infty}^x f(u) du$, so we have to at least discuss if the $-\infty$ affects things. In this case, and in general the answer is <em>no</em>. For any $x$ we can find $M < x$ so that:</p>$$F(x) = \int_{-\infty}^M f(u) du + \int_M^x f(u) du$$<p>. The first part is a constant, so will have derivative $0$, the second will have derivative $f(x)$, if the derivative exists (and it will exist at $x$ if the derivative is continuous in a neighborhood of $x$).</p><p>Finally, at $x=0$ we have an issue, as $F'(0)$ does not exist. The left limit of the secant line approximation is $0$, the right limit of the secant line approximation is $1$. So, we can take $f(x) = e^{-x}$ for $x > 0$ and $0$ otherwise, noting that redefining $f(x)$ at a point will not effect the integral as long as the point is finite.</p><h2>Questions</h2><h6>Question</h6><p>Is $f(x) = 1/x^{100}$ integrable around 0</p><form name="WeaveQuestion" data-id="jTGtavia" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_jTGtavia" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_jTGtavia" value="2"><div class="markdown"><p>No</p>
</div>
</label>
</div>
<div id="jTGtavia_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_jTGtavia']").on("change", function() {
correct = this.value == 2;
if(correct) {
$("#jTGtavia_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#jTGtavia_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
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<h6>Question</h6><p>Is $f(x) = 1/x^{1/3}$ integrable around 0</p><form name="WeaveQuestion" data-id="FfUzSQlw" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_FfUzSQlw" value="1"><div class="markdown"><p>Yes</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_FfUzSQlw" value="2"><div class="markdown"><p>No</p>
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</div>
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} else {
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<h6>Question</h6><p>Is $f(x) = x\cdot\log(x)$ integrable on $[1,\infty)$?</p><form name="WeaveQuestion" data-id="B96HYMvp" data-controltype="radio">
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<input type="radio" name="radio_B96HYMvp" value="2"><div class="markdown"><p>No</p>
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} else {
$("#B96HYMvp_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>Is $f(x) = \log(x)/ x$ integrable on $[1,\infty)$?</p><form name="WeaveQuestion" data-id="CXeqxi8d" data-controltype="radio">
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<div class="radio">
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<input type="radio" name="radio_CXeqxi8d" value="1"><div class="markdown"><p>Yes</p>
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</div>
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<input type="radio" name="radio_CXeqxi8d" value="2"><div class="markdown"><p>No</p>
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<h6>Question</h6><p>Is $f(x) = \log(x)$ integrable on $[1,\infty)$?</p><form name="WeaveQuestion" data-id="BwwUamMe" data-controltype="radio">
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<input type="radio" name="radio_BwwUamMe" value="1"><div class="markdown"><p>Yes</p>
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<div class="radio">
<label>
<input type="radio" name="radio_BwwUamMe" value="2"><div class="markdown"><p>No</p>
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<h6>Question</h6><p>Compute the integral $\int_0^\infty 1/(1+x^2) dx$.</p><form name='WeaveQuestion' data-id='cebXQeXZ' data-controltype='numeric'>
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<div class='controls'>
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</div>
</div>
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<h6>Question</h6><p>Compute the the integral $\int_1^\infty \log(x)/x^2 dx$.</p><form name='WeaveQuestion' data-id='omC6upDS' data-controltype='numeric'>
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<div class='controls'>
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</div>
</div>
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}
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</script>
<h6>Question</h6><p>Compute the integral $\int_0^2 (x-1)^{2/3} dx$.</p><form name='WeaveQuestion' data-id='KZyCri1W' data-controltype='numeric'>
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} else {
$('#KZyCri1W_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
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});
</script>
<h6>Question</h6><p>From the relationship that if $0 \leq f(x) \leq g(x)$ then $\int_a^b f(x) dx \leq \int_a^b g(x) dx$ it can be deduced that</p><ul>
<li>if $\int_a^\infty f(x) dx$ diverges, the so does $\int_a^\infty g(x) dx$.</li>
<li>if $\int_a^\infty g(x) dx$ converges, the so does $\int_a^\infty g(x) dx$.</li>
</ul><p>Let $f(x) = \lvert \sin(x)/x^2 \rvert$.</p><p>What can you say about $\int_1^\infty f(x) dx$, as $f(x) \leq 1/x^2$ on $[1, \infty)$?</p><form name="WeaveQuestion" data-id="EMub12SC" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_EMub12SC" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_EMub12SC" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_EMub12SC" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="EMub12SC_message"></div>
</div>
</form>
<script text="text/javascript">
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} else {
$("#EMub12SC_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<hr /><p>Let $f(x) = \lvert \sin(x) \rvert / x$.</p><p>What can you say about $\int_1^\infty f(x) dx$, as $f(x) \leq 1/x$ on $[1, \infty)$?</p><form name="WeaveQuestion" data-id="sLhLxbra" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_sLhLxbra" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_sLhLxbra" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_sLhLxbra" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="sLhLxbra_message"></div>
</div>
</form>
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} else {
$("#sLhLxbra_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<hr /><p>Let $f(x) = 1/\sqrt{x^2 - 1}$. What can you say about $\int_1^\infty f(x) dx$, as $f(x) \geq 1/x$ on $[1, \infty)$?</p><form name="WeaveQuestion" data-id="OyaxVp4w" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_OyaxVp4w" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_OyaxVp4w" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_OyaxVp4w" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="OyaxVp4w_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_OyaxVp4w']").on("change", function() {
correct = this.value == 2;
if(correct) {
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<hr /><p>Let $f(x) = 1 + 4x^2$. What can you say about $\int_1^\infty f(x) dx$, as $f(x) \leq 1/x^2$ on $[1, \infty)$?</p><form name="WeaveQuestion" data-id="kC2DVjuA" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_kC2DVjuA" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_kC2DVjuA" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_kC2DVjuA" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="kC2DVjuA_message"></div>
</div>
</form>
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$("#kC2DVjuA_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
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});
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<hr /><p>Let $f(x) = \lvert \sin(x)^{10}\rvert/e^x$. What can you say about $\int_1^\infty f(x) dx$, as $f(x) \leq e^{-x}$ on $[1, \infty)$?</p><form name="WeaveQuestion" data-id="gHMnxG72" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_gHMnxG72" value="1"><div class="markdown"><p>It is convergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_gHMnxG72" value="2"><div class="markdown"><p>It is divergent</p>
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_gHMnxG72" value="3"><div class="markdown"><p>Can't say</p>
</div>
</label>
</div>
<div id="gHMnxG72_message"></div>
</div>
</form>
<script text="text/javascript">
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$("#gHMnxG72_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>The difference between "blowing up" at $0$ versus being integrable at $\infty$ can be seen to be related through the $u$-substitution $u=1/x$. With this $u$-substitution, what becomes of $\int_0^1 x^{-2/3} dx$?</p><form name="WeaveQuestion" data-id="ANwO4wkN" data-controltype="radio">
<div class="form-group ">
<div class="radio">
<label>
<input type="radio" name="radio_ANwO4wkN" value="1"><div class="markdown">$\int_1^\infty u^{2/3}/u^2 \cdot du$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_ANwO4wkN" value="2"><div class="markdown">$\int_0^\infty 1/u \cdot du$
</div>
</label>
</div>
<div class="radio">
<label>
<input type="radio" name="radio_ANwO4wkN" value="3"><div class="markdown">$\int_0^1 u^{2/3} \cdot du$
</div>
</label>
</div>
<div id="ANwO4wkN_message"></div>
</div>
</form>
<script text="text/javascript">
$("input:radio[name='radio_ANwO4wkN']").on("change", function() {
correct = this.value == 1;
if(correct) {
$("#ANwO4wkN_message").html("<div class='alert alert-success'><span class='glyphicon glyphicon-thumbs-up'> Correct</span></div>");
} else {
$("#ANwO4wkN_message").html("<div class='alert alert-warning'><span class='glyphicon glyphicon-thumbs-down'> Incorrect</span></div>");
}
});
</script>
<h6>Question</h6><p>The antiderivative of $f(x) = 1/\pi \cdot 1/\sqrt{x(1-x)}$ is $F(x)=(2/\pi)\cdot \sin^{-1}(\sqrt{x})$.</p><p>Find $\int_0^1 f(x) dx$.</p><form name='WeaveQuestion' data-id='ULG3Jrxa' data-controltype='numeric'>
<div class='form-group '>
<div class='controls'>
<div class="input-group">
<input id="ULG3Jrxa" type="number" class="form-control">
</div>
<div id='ULG3Jrxa_message'></div>
</div>
</div>
</form>
<script text='text/javascript'>
$('#ULG3Jrxa').on('change', function() {
correct = Math.abs(this.value - 0.9999999921866226) <= 0.001;
if(correct) {
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$('#ULG3Jrxa_message').html('<div class="alert alert-danger"><span class="glyphicon glyphicon-thumbs-down"> Incorrect</span></div>');
}
});
</script>
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