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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0
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1answer
14 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
2
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1answer
50 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
1
vote
1answer
32 views

Improper Integral - Multiple Choice Problem - $I$

Let $f$ be a function defined $\forall~ x\geq 1$.Let $n$ denote a positive integer and let $I_n$ denote the integral $\int_1^nf(x)dx$ which is always assumed to exist. Which of the following ...
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3answers
39 views

Divergence/convergence of an integral

I am told that the following integral converges for $1<n<3$. $$ \int_{-\infty}^{+\infty} (1-e^{ix}) |x|^{-n} dx $$ I am a bit baffled. Anyone with a clue or where to start with this in order to ...
0
votes
1answer
54 views

Integral $\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$

How to evaluate: $$\int_0^\infty e^{-x/2}x\log(1+kx^2)\,dx$$ Basically am evaluating value of $\log(1+c\chi^2)$ where $\chi^2$ is $\chi$-squared distributed
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3answers
77 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
3
votes
5answers
90 views

Explain why the integral $\int_{-\infty}^\infty x \,dx$ does not exist

Why is it that $$\int_{-\infty}^\infty x \,dx$$ does not exist, but $$\lim_{N \to \infty} \int_{-N}^{N} x\,dx$$ does exist? I was thinking that it involves the fact that in the second case, the ...
2
votes
2answers
27 views

Improper integral calculation - limit at infinity

Will you please help me prove the following limit is zero ? $$\lim_{x \to \infty} \int_0^{\infty} \frac{1-e^{-u^4}}{u^2} \cos(x u) du. $$ Thanks in advance
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4answers
73 views

Convergence of $\int_0 ^\infty \frac {dx}{\sqrt {1+x^3}}$

Convergence of $\int_0 ^\infty \dfrac {dx}{\sqrt {1+x^3}}$ Attempt: $\lim_{x \rightarrow \infty} \dfrac {x^{\frac{3}{2}}}{\sqrt {1+x^3}} =1$ Hence, $\dfrac {1}{x^{\frac{3}{2}}}$ and $\dfrac ...
31
votes
4answers
802 views

What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?

/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Worlfram alpha gives an indefinite form in terms of the logarithmic integral ...
2
votes
2answers
70 views

Unusual integral

I got a clock as a gift recently. It has a very novel face in that the hour positions are given by a complex formula. For the most part, I have been able to verify the calculations presented as ...
3
votes
5answers
80 views

How to show $\int_0^\infty\frac{dx}{x\sqrt{1-x^2}}=\pi/2$

How to show that $$\int_0^\infty\frac{dx}{x\sqrt{1-x^2}}=\frac{\pi}{2}$$ The problem is that I don't know what is $$\lim\limits_{x\to\infty}{\mathrm{arcsec}\ x}$$
0
votes
0answers
18 views

Simple proof for a continuous-time linear system and impulse $\delta$?

From Schaum's Outlines of Signals & Systems: Let's work with continuous-time signals. Let $T$ be a linear time-invariant system (LTI). Input $x(t)$ can be expressed as $x(t) = ...
0
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0answers
45 views

Problem about limit of an integral

I came across this question while doing some exercises on integrals, and I was wondering if I could get some help. a) Show that for $n < -1$, $\int_1^N x^n dx$ converges as $N \to\infty$, and for ...
0
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1answer
26 views

Differentiating an Integral

Does anyone know any general approach for something like this: $$ \frac{d}{dx}\int_{-\infty}^{x}f(x,u)du\qquad\text{or}\qquad\frac{d}{dx}\int_{x}^{\infty}f(x,u)du\qquad $$ Basically, I'm trying to ...
2
votes
2answers
106 views

How to compute this integral involving sech?

Does anybody know how to so solve this integral: $$ \int\limits_{-\infty}^{+\infty}\text{sech}(t-h)\,e^{ikt/\epsilon}\,\mathrm{d}t $$ and where $h\in \mathbb{R}$ is a constant, $k\in\mathbb{Z}$ and ...
4
votes
1answer
169 views

How to calculate the two integrals?

How calculate the following integrals $$I=\int_0^{\pi/2}x^2\frac{(1-\tan\,x)\sin4x}{\sqrt{\tan x}}dx$$ $$K=\int_0^{\pi/2}x^2\frac{(1+\tan\,x)\sin4x}{\sqrt{\tan x}}dx$$ EDIT It seems that one ...
1
vote
3answers
77 views

Convergence of $\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx $

I would like to know if the improper integral $$\int^\infty_0 \frac{e^{-\sqrt x}}{1+x}dx \qquad (1)$$ is convergent or not. I tried substitution and integration by parts but got no simplification. So, ...
2
votes
2answers
58 views

Convergence of $\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$

Test the convergence of $$\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$$ Attempt: For sufficiently large $x$, we have $e^{-\sqrt x} > e ^{- x}$. I also tried solving the integral by By Parts ...
2
votes
1answer
259 views

Improper integral (is it convergent?) (v 2.0)

Earlier today I asked about this question: Improper integral (is it convergent?) where the integral fortunately seems to be convergent. So we have that given $\alpha\in (-1/2,0)$ there is a $\gamma ...
3
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1answer
35 views

Computing with Lebesgue integrals

This problem comes from Royden's Real Analysis, 4th ed., pg 84, #19: For a number $\alpha$, define $f(x)=x^\alpha$ for $0<x\le 1$ and $f(0)=0$. Compute $\int_0^1 f$. MY WORK: I know ...
1
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1answer
76 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
2
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2answers
54 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
0
votes
1answer
53 views

Convergence of an integral $\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$

For what $\alpha\in\Bbb{R}$ does $\displaystyle\int_0^\infty\frac{dt}{1+t^\alpha\sin^2(t)}$ converge ? The $0$ bound doesn't seem to be much of a problem, but I don't see how to deal with the ...
1
vote
1answer
45 views

Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
3
votes
2answers
76 views

Why does not $\int_{-\infty}^\infty x\,\mathrm{d}x$ converge?

It seems natural that it should converge, because for any $A\in\mathbb{R}$, $$\int_{-A}^A x\,\mathrm{d}x=\int_{-A}^0 x\,\mathrm{d}x+\int_0^A ...
2
votes
1answer
73 views

Improper integral: $\,\frac{1}{\pi}\int^\infty_0 \frac{\sqrt{x}}{1+x}e^{-xt}\,dx$

Good evening! How could one evaluate the following integral $$\frac{1}{\pi}\int^\infty_0 \frac{\sqrt{x}}{1+x}e^{-xt}\,dx$$ I have tried the substitution $x\equiv x^2$ but still I could not manage to ...
2
votes
2answers
65 views

Improper integral (is it convergent?)

I would like to either prove or disapprove the following: Let $\alpha\in (-1/2,0)$ be given. Then we can find $\gamma \in (1,2)$ such that $$\int_0^1 \int_0^{u} ...
0
votes
1answer
25 views

Compute an integral explicit (or explicit upperbound? Possible?)

I would just like to know if there is an explicit formula/computation for this integral: $$\int_{a}^1 \left(|x-a|^{\alpha} - |x-b|^{\alpha} \right)^2 dx$$ where $\alpha\in (-1/2,0)$ so the integral ...
3
votes
2answers
56 views

Asymptotic form of an integral

I would like to find an asymptotic form of the following integral when $s \to \infty$ ($s$ and $w$ are positive) \begin{equation} \int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs} \end{equation} I ...
2
votes
1answer
88 views

Evaluating $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ using complex analysis

Again, improper integrals involving $\ln(1+x^2)$ I am trying to get a result for the integral $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ - asked above link- using some complex ...
2
votes
1answer
33 views

Integral on $\mathbb{R}^d$

This is probably a simple question, but I don't have tons of experience integrating on $\mathbb{R}^d$ for arbitrary $d$. I'd like to compute the following integral $$ \int_{\mathbb{R}^d} ...
4
votes
2answers
216 views

About the Beta function : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$.

Find the value of : $\text{B}\left(\frac{4}{3},\frac{2}{3}\right)$, where $\text{B}(x,y)$ is the Beta function. Why do I need this ? Because I want to calculate : $$ \int\limits_{ - \infty }^\infty ...
7
votes
3answers
176 views

Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$

Evaluate $$\displaystyle\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$$ How do I evaluate this integral? I know that the result is $0$, but I don't know how to obtain this. Wolfram|Alpha ...
17
votes
5answers
1k views

Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...
10
votes
5answers
28k views

Integral: $\int_{-\infty}^{\infty} x^2 e^{-x^2}\mathrm dx$

I don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\int_{-\infty}^\infty x^2 ...
4
votes
6answers
215 views

Prove that $\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}$

Good evening everyone, how can I prove that $$\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}\;?$$ Well, I know that $\displaystyle\frac{1}{x^4+x^2+1} $ is an even function ...
1
vote
2answers
35 views

Studying the divergence or otherwise of an improper integral

I'm supposed to study the following improper integral : $$\int^{1/2}_{-\infty}\frac{1}{1-x^{1/3}} dx$$ This could be an exercise out of 5 in our final exam paper, so I reckon there's a fast way to do ...
0
votes
1answer
41 views

what is being asked here?

I fail to see how this can be achieved: Define the improper integral (of a non-blocked function) as a limit, and calculate or prove that the integral diverges; $\large \int_0^1 \frac{dx}{ ...
3
votes
2answers
73 views

The value of $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$

Using the fact $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos nx dx=0$ ,find the value of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$$ I tried through integrating by parts , ...
13
votes
4answers
342 views

Prove that $\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$

Prove that (please) $$\int_0^1\frac{\ln(1-x)\ln^2x}{x-1}dx=\frac{\pi^4}{180}$$ I've tried using Taylor series and I ended up with $$-\sum_{m=0}^\infty\sum_{n=1}^\infty\frac{2}{n(m+n+1)^3}$$ I am ...
2
votes
3answers
140 views

Help solving an improper integral

I need to solve an improper integral which is: $$ \int_1^\infty \frac{2}{4{x^2}-1}dx $$ i was trying to solve it using simple substitution but cannot seem to figure it out, i tried a website to ...
0
votes
0answers
77 views

Derivative of a double integral (applying Leibniz rule)

I would like to differentiate the following expected value function with respect to parameter $\beta$: $$F(\xi_1,\xi_2) ...
3
votes
4answers
66 views

$\int_0^{\infty}y^2e^{-y} dy$

To calculate $\displaystyle \int_0^{\infty}y^2e^{-y} dy$ =$\displaystyle -y^2e^{-y}-2ye^{-y}-2e^{-y}|_o^{\infty}$ This should fail at $\infty$, but the answer is given as 2. Seems like $e^{-y}$ ...
15
votes
5answers
643 views

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \begin{equation} \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
4
votes
1answer
65 views

Does the integral $\int_0^\infty \sin(2x^4) \, dx$ converge absolutely/conditionally?

Does the integral $$\int_0^\infty \sin(2x^4) \, dx$$ converge absolutely/conditionally? I tried to evaluate $\int_0^b \sin(2x^4) dx$ by integrating by parts twice and got something relatively ...
4
votes
3answers
451 views

Integrals $ \int_0^1 \log x \mathrm dx $,$\int_2^\infty \frac{\log x}{x} \mathrm dx $,$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$

I don't get how we're supposed to use analysis to calculate things like: a) $$ \int_0^1 \log x \mathrm dx $$ b) $$\int_2^\infty \frac{\log x}{x} \mathrm dx $$ c) $$\int_0^\infty \frac{1}{1+x^2} ...
2
votes
4answers
258 views

Value of an unbounded definite integral

Evaluate the integration : $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(2x^{2}+2xy+2y^{2})}dxdy$$ The function is even about $x$ & $y$. So we can write, ...
10
votes
3answers
233 views

How to prove $\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$?

How to prove the following result? $$\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$$ For my part no idea?