Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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1
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2answers
23 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
1
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0answers
19 views

Continuity of improper integrals

There is a theorem saying that if $f:[a,b]\to \mathbb R$ is integrable on $[a,b]$, then $F(x):=\int_{a}^{x}f(t)dt$ is continuous on $[a,b], x \in [a,b]$. Is there an analogous theorem of the kind: ...
3
votes
3answers
62 views

Convergence of doubly infinite improper integral for odd functions.

I was working on this integral: $$\int_{-\infty}^{+\infty} \frac{x \, dx}{1+x^2}$$ Calculations shows that the limits DNE, and therefore the integral diverge. I used Mathematica and found the same ...
-1
votes
0answers
34 views

$\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$

Could someone kindly evaluate $\int_{1}^{\infty} \frac{\omega^2-x^2}{(\omega^2+x^2)^2}(x^2-1)^{-(2/3)}dx$ for me? Cheers, Allen
1
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0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
4
votes
7answers
128 views

Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $

What is an appropriate method to approximate $$I=\int_0^\infty e^{-t^2 /100} \sin \pi t \ dt?$$ This is for a Physics problem, but in fact I need this in general, as my professor and book taught us ...
1
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1answer
30 views

Integration of Bessel functions:Finding a suitable contour

I have below function to integrate; $$\int_{0}^{\infty} \frac{J_{0}(ax)x^3}{k^2-x^2} dx$$ here $a,k$ are constants. Since this is an odd function, I am not allowed to extend the limits from negative ...
0
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0answers
49 views

Contour integration from zero to infinity

When solving an improper integration from $0$ to $\infty$ which involves an even function, the integration limits can be extended from $-\infty$ to $\infty$. For example consider even function $f(x)$; ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
0
votes
1answer
72 views

Definite integration by induction

$U_n= \int\frac{x^n}{((x(1-x))^{0.5}}$ where $0<x<1$ Prove that $2nU_n=(2n-1)U_{n-1}$ My work I did $U_0=\pi, u_1=\pi/2$ so its true for $n=1$
2
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5answers
55 views

How to calculate: $ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $

How do I calculate the follwing Limit: $$ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $$ I have been solving an exam from my University's collection of ...
2
votes
1answer
58 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
8
votes
3answers
87 views

Is there a direct method for evaluating this integral: $\int_{0}^{2\pi}\ln^2(2\sin(\frac{x}{2}))dx$?

I stumbled upon this integral while attempting to evaluate $\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}$. I started with the series $-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^n}{n}$, replaced z with ...
1
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0answers
47 views

Integrate $\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c}dx$ where $P$ is a polynomial of degree $6$

From a physics problem I'm interested by a closed form of this integral : $$\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c} dx$$ where $P(x) = \lambda_6 x^6 + ... + \lambda_0$ I ...
2
votes
1answer
44 views

Finding the value of an integral containing $(\ln x)^2$ in the denominator

While reviewing (as an instructor/test editor) a second semester calculus exam, I came across the following problem: Find the volume of the solid created by revolving around the $x$-axis the area ...
1
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3answers
73 views

What is wrong with my method of computing $\int_0^{\infty}x^2 e^{-x^2} \space dx$

I want to check if the following improper integral converges or diverges: $$\int_0^{\infty} x^2e^{-x^2}\space dx$$ Rewriting the integrand: $$\int_0^{\infty}-\frac{1}{2}x(-2x e^{-x^2}) \space dx$$ ...
4
votes
1answer
88 views

Exchanging the order of integration in $ \int_0^\infty \int_{-\infty}^\infty \sin(x^2)x e^{-t^2 x^2} dt dx$?

For context, this gives one way to evaluate the Fresnel sine integral at infinity. The problem I'm running into is $$ \int_0^\infty \left[ \int_{-\infty}^\infty \vert\sin(x^2)x e^{-t^2 x^2}\vert dt ...
3
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1answer
113 views

Help with a limit of an integral

I'm not sure how to handle limits and integral and I would like some help with the following one: let $f:[0,\infty)\rightarrow \Bbb{R}$ be a continuous and bounded function, show that ...
6
votes
3answers
93 views

Integrating $\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}$ with respect to $\theta$

I'm having some issues with the following integral $$\int_{\frac{-\pi}{2}}^\frac{\pi}{2}\frac{\sec^2\theta}{1+\tan^2\theta \cos^2(2\alpha)}d\theta$$ My attempt is as follows, substitute ...
10
votes
2answers
129 views

Computing $\lim_{\epsilon \rightarrow 0} \int_0^\infty \frac{\sin x}{x} \arctan{\frac{x}{\epsilon}}dx$

I'm not exactly sure how to get started computing the limit of the improper Riemann integral $$\lim_{\epsilon \rightarrow 0} \int_0^\infty \frac{\sin x}{x} \arctan\left(\frac{x}{\epsilon}\right)dx.$$ ...
-1
votes
0answers
57 views

Any comment on an integral [closed]

Any solution/approximation for $$ \int_{1}^{\infty} \! r \exp\left\{- \left( a r +b r^{-c} \right) \right\} \, \mathrm{d}r \:,$$ where $a \geq 0$, $b>0$, and $c>2$ are constant?
2
votes
0answers
98 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
0
votes
0answers
82 views

How can I resolve this improper integral?

I would like to resolve this integral numerically . However, I'm not sure about the best way to do it because it is an improper integral: $$ ...
0
votes
3answers
80 views

what is the integral $\int_{0}^1 \sqrt{(1+(1/x^2))}dx$

Can this integral can be evaluated? When I attempted it I got an answer of infinity. Maybe I don't know the correct method of solving this integration problem. Please provide the correct answer with ...
1
vote
0answers
21 views

Leibniz rule for probability distribution with infinite support.

Let $f$ be the pdf of a non-negative random variable $X$ with finite moments of all orders, i.e. $E[X^n]<+\infty$ for all $n \in \mathbb N$. May I apply Leibniz's rule and infer that $$\frac{d}{d ...
2
votes
2answers
64 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
1
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1answer
35 views

Improper integral existence exercise

$$\int _0^1\frac{\ln (1+\sqrt{x})}{\sin(x)} \, dx\:$$ So the singularity point is at $0$, so we`ll use this test: $$\lim _{x\to 0}\frac{\frac{\ln(1+\sqrt{x})}{\sin(x)}}{\frac{1}{\sqrt{x}}}=\lim_{x\to ...
0
votes
0answers
21 views

Compute asymptotic expansion of an integral along the unit circle

I want to compute the asymptotic expansion of the following integral with $t\rightarrow +\infty$ $\int_C\dfrac{(1+u)^{t+4}}{u^5}du$ where $C$ is the unit circle. I really appreciate your help. By ...
1
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0answers
38 views

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by ...
4
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1answer
44 views

Find the Principal Value of the integral $\int_{-\infty}^\infty \frac{x sin(x)}{x^2+2x+2}dx$

This problem comes from a preliminary exam from 2009 "Find the Principal Value of the integral $$\int_{-\infty}^\infty \frac{x \sin(x)}{x^2+2x+2}dx"$$ My attempt at solution: Letting $f(z)=\frac{z ...
2
votes
2answers
51 views

Calculate integral $\int_0^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$ for $b>0$ and any $a \in \mathbb{R}$

i am working on following task: Choose any nonzero $a \in \mathbb{R}$ so the integral converges and for a given $b > 0$ compute $\int_t^\infty e^{-x} (e^{-\frac a b x} - 1)^{b} dx$. I am looking ...
6
votes
3answers
152 views

How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$.

This $$\ \int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx \text{ where } a,b > 0. $$ is a problem that showed up on a GRE practice test. I believe you're supposed to use complex ...
2
votes
0answers
27 views

The proof of the integral test using the contradiction method.

I am currently writing a short note about the proof techniques. I found a random theorem and wanted to write a proof by contradiction as an example. The theorem says The integral ...
2
votes
2answers
68 views

Evaluate the integral by type1 or type 2

Evaluate $\displaystyle\int_{0}^{2} \int_{0}^{\log(x)}(x-1)\sqrt{1+e^y}\,dy\,dx$. I have tried integration by substitution but can't connect to type 1 or type 2. Any help.
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0answers
45 views

Improper integral $\int \limits _0 ^{\infty} \frac {x^{n+a-1} \Bbb e ^{-x}} {\left[ 1+k \, \gamma(a,x) \right] ^{\theta+1}} \Bbb d x$

Can anyone help me with the following integal: $$\int \limits _0 ^\infty \frac {x^{n+a-1} \, \Bbb e ^{-x}} {\left[ 1+k \, \gamma(a,x) \right] ^{\theta +1}}\, \Bbb d x,$$ where $n \in \mathbb{N},$ ...
1
vote
2answers
76 views

Estimating the value of an improper integral numerically

My question is how can I estimate the value of an improper integral from $[0,\infty)$ if I only have a programming routine that gives me the function evaluated at 100 data points, or 100 values of ...
0
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1answer
49 views

Identities for integral expressions like $\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$

I have some fairly complicated integrals to work with, and sometimes to divide by each other. $$\frac{\int_0^\infty a(x)f(x)dx}{\int_0^\infty a(x)g(x)dx}$$ Are there any identities that can be used ...
1
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2answers
59 views

Prove convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$

Prove the convergence of $\int_1^\infty \frac 1 {x(\sqrt x + 1)} dx$ This was a question on an exam. I needed to prove that the above integral converges using the comparison test. I thought about ...
1
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2answers
21 views

Space of all improper Riemann-integrable functions not closed under products and other operations

If $R[a,b]$ denotes the space of all Riemann-integrable functions in the closed interval $[a,b]$, then this space is closed under taking linear combinations, product of functions, powers of functions ...
1
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2answers
50 views

Improper integrals - Showing convergence.

1)Show that for all $n\in\mathbb{N}$ the following is true: $\int_{\pi}^{n\pi}|\frac{\sin(x)}{x}|dx\geq C\cdot \sum_{k=1}^{n-1}\frac{1}{k+1}$ for a constant $C>0$ and conclude that ...
2
votes
3answers
110 views

$\int_1^\infty 1/\sqrt{1+e^x}dx$

Using the comparison test I am supposed to figure out whether this integral converges or diverges. what other function should I use? Also, the inequality stating that $1/\sqrt{e^x+1}$ is larger or ...
2
votes
2answers
51 views

An improper integral and its convegence

I have an integral $$I(\gamma)=\int\int d^3 \mathbf{r} \, d^3 \mathbf{r}' \frac{1}{|\mathbf{r}-\mathbf{r}'|+\gamma}$$ were $\gamma$ is a positive number, $\mathbf{r},\mathbf{r}' \in \mathbb{R}^3$, ...
0
votes
3answers
46 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...
1
vote
2answers
42 views

Struggling with the integrability of $\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$

I know quite a lot tools to determine the integrability of functions, but in this case I really don't know where to start: $$\int_{\frac{\pi}{2}}^{\pi}(\tan(x))^{\frac{1}{3}}\text{d}x$$
2
votes
1answer
95 views

Investigate the convergence of $\int _0^\infty \frac{\sin x^2}{x} \ dx$

Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely? I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ ...
0
votes
2answers
38 views

How can I solve this integral with the comparison theorem?

I have an integral that I am not sure how to solve with the comparison theorem to see if it is divergent or convergent. $$\int_1^\infty\frac{e^{-2x}}{\sqrt{x+16}}\;dx$$ How can I solve this with ...
1
vote
2answers
61 views

Conditional convergence and Riemann's series theorem

There are tests to determine whether an integral or sum is convergent. There are test to determine whether an integral or sum is absolutely convergent. An integral or series is said to be $\mathbf ...
2
votes
2answers
141 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$ [duplicate]

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
1
vote
1answer
38 views

Integral Test question

So this is the problem: http://postimg.org/image/5g815zgk5/ I am getting $\lim_{b\to\infty} 2\sec^{-1}(2b) - 2\sec^{-1}2$ Now what? What do I do with $\sec^{-1}(2b)$? What happens to a trig function ...
0
votes
2answers
58 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...