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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1answer
87 views

From $\int_{1}^{+\infty} f(x)dx$ is convegent , we can say $\lim_{x\to +\infty}f(x)=0$?

From $\int\limits_{1}^{+\infty} f(x)dx$ is convegent , we can say $\lim_{x\to +\infty}f(x)=0$? I think it's true . Because some example for me it. ...
2
votes
1answer
154 views

Improper Multiple Integral

I am trying to solve the following exam problem: Let $s$ be a real number. Find the condition under which the improper integral $$I:=\iint_{\mathbb R^2} \frac{dxdy}{(x^2-xy+y + 1)^s}$$ converges, ...
7
votes
4answers
422 views

Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$

Compute improper integral : $\displaystyle I=\int\limits_{0}^{+\infty}\dfrac{\arctan \pi x-\arctan x}{x}dx$.
2
votes
1answer
87 views

How to prove : $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent

How to prove : If $\int\limits_{1}^{+\infty} x.f(x)dx$ is convergent then $\int\limits_{1}^{+\infty} .f(x)dx$ is too convergent.
3
votes
3answers
289 views

Does $\displaystyle\int_{-\infty}^{\infty} e^{-\sqrt{|x|}}\,\mathrm{d}x$ converge?

I would like to find out if this integral converges: $\displaystyle\int_{-\infty}^{\infty} e^{-\sqrt{|x|}}\,\mathrm{d}x$ Since this is a symmetric function I figured I could focus on only one side ...
14
votes
3answers
333 views

Does $\int\limits_0^\infty \cos(x^3 -x) \, \mathrm dx$ converge?

Does $\displaystyle\int_0^\infty \cos(x^3 -x) \, \mathrm dx$ converge? What is the standard method of checking convergence of this kind of improper integrals?
0
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2answers
57 views

how to find this type of definite double integral?

could any one tell me how to find this type of definite double integral? $$\int_{0}^{\infty}\int_{x}^{\infty}{e^{{-y\over2}}\over y}dydx$$ Thank you.
14
votes
3answers
403 views

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$

If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x)dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left( x - ...
5
votes
1answer
155 views

Prove: $\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$

I want to solve the following integral: $$\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$$ Have not tried it yet, but it may be tough. All I know is that ...
2
votes
1answer
108 views

Integral of function with a pretty long name

I am looking for a compact result of this integral: $$\int_R^\infty k_n(x) \, dx,$$ where $k_n$ is the modified spherical Bessel function of the second kind (explanation to this function). ...
0
votes
1answer
70 views

Is my reasoning correct about the convergence of this integral?

The integral is $\int_1^\infty\frac{\sin{x}}{x}dx$. I know that this integral converges, but I'm wondering if this is valid way to prove it. This function, if its domain is limited to ...
36
votes
1answer
971 views

To evaluate $\int_0^{+\infty} \frac{\;dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$ ...
6
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1answer
195 views

how to calculate this complementary Bessel function?

I am trying to calculate this complementary Bessel function $$\Psi(a,b,\gamma)=\int_0^\infty\Phi({a\over \sqrt{u}}+b\sqrt{u}){u^{\gamma-1}e^{-u}\over \Gamma(\gamma)}du$$ where $\Phi$ is the standard ...
4
votes
0answers
90 views

Generalized sine integral $ \int_0^\infty \sin^m(x^n)/x^p\,\mathrm dx$

I have seen that both the integrals $$ \color{black}{ \int_0^\infty \operatorname{sinc}(x^n)\,\mathrm{d}x = \frac{1}{n-1} \cos\left( \frac{\pi}{2n}\right)\Gamma ...
1
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3answers
97 views

Convergence of Improper Integral: $\int_{e^2}^\infty {dx\over x\log\log x}$

Test the convergence of the following integral$$\int_{e^2}^\infty {dx\over x\log\log x}$$ I understand that the problem is only at $\infty$ how to proceed ?
1
vote
5answers
125 views

test convergence of improper integrals 4

Test the convergence of improper integrals : $$\int_1^2{\sqrt x\over \log x}dx$$ I basically have no idea how to approach a problem in which log appears. Need some hint on solving this type of ...
1
vote
2answers
135 views

How to integrate these integrals?

This question was a question in an exam years ago. Find the values of the following integrals. (i) $$\int_\Gamma\dfrac{xdy-ydx}{x^2+y^2},$$ where $\Gamma$ is the curve $x=t\cos t$, $y=t^2\sin ...
3
votes
2answers
111 views

Improper integral that converges for all $x$ in $ \mathbb{R}$

Let $f(x)$ be defined by the improper integral: $$f(x)= \int_{0}^{\infty} \cos\left(\frac{t^3}{3} + \frac{x^2 t^2}{2} + xt\right)dt.$$ Show that this improper integral converges for all $x ...
2
votes
4answers
66 views

test for convergence of improper integral1

$$\int_0^1 {x^n\log x\over(1+x)^2} \, dx$$ I tried something using practical test, but not much progress. I see that the integral becomes improper for $x=0$, May be we need to apply the Practical ...
1
vote
1answer
49 views

Convergence of Improper Integrals2

Test the convergence of $$\int_0^{\pi/2}\frac{\sin x}{x^n}\,dx$$ I tried doing it by comparison test by taking $\phi(x)=\dfrac{1}{x^n}$. Then $$\lim_{n\rightarrow ...
4
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1answer
83 views

Estimate a upper bound of an infinite series.

Assume $a>0$ and $a_n \geq 0$. how to verify that $$\sum_{n=1}^{\infty}\frac{a_n}{(a+S_n)^{3/2}}\leq \int_0^{\infty}\frac{1}{(a+x)^{3/2}}\mathrm{d}x$$ where $S_n = a_1+a_2+\cdots+a_n$ Thanks very ...
2
votes
1answer
119 views

Oscillating integral

I want to calculate $$ \int _0^\infty e^{-iyx}\sqrt{x(x+2)}\, dx $$ in the sense of distributions, at least for $y\ne 0$. Now, I happen to know the following integral representation for the modified ...
4
votes
1answer
142 views

When are the following multiple improper integrals convergent?

This is a question from a past exam. For which $p, q\in \mathbb R$ do the following improper integrals converge? ...
4
votes
6answers
228 views

A problem about the convergence of an improper integral

Let $f:\mathbb R\longrightarrow\mathbb R$ be a function with $$f(x)=\frac{1}{3}\int_x^{x+3} e^{-t^2}dt$$ and consider $g(x)=x^nf(x)$ where $n\in\mathbb Z$. I have to discuss the convergence of ...
8
votes
2answers
165 views

Stuck on the integral $\int_0^{\infty}\frac{2+7\mathrm{cos}(x^\pi-e)-7\mathrm{sin}(1+x^8)}{1+x^2} \mathrm{d}x$

Does anyone have any advice on how to evaluate the following integral? $$\int_0^{\infty}\frac{2+7\mathrm{cos}(x^\pi-e)-7\mathrm{sin}(1+x^8)}{1+x^2} \mathrm{d}x$$ It looks like it converges, but I ...
2
votes
0answers
52 views

calculating this improper integral [duplicate]

I have problems when calculating this improper integral $$\int_{-\infty}^{\infty}\frac{e^{-x^2}}{1+x^2}e^{\frac{2x^2t}{1+t}}dx~= ~?~~~|t|<1$$ assistance would be helpful.
3
votes
1answer
136 views

improper integral calculation

I do not know how to solve this integral and then expand the result in powers of $t$ $$(1-t)^{-1/2}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{2x^2t}{1+t}\right)}{1+x^2}dx=?,~~~~|t|<1.$$ any ...
1
vote
1answer
1k views

A method for calculating this integral hermite polynomials

I need proof this, $\int_{-\infty}^{\infty}e^{-x^2}H_n^{2}(x)x^2dx=2^nn!\sqrt{\pi}(n+\frac{1}{2})$ This is the idea: Multiply ...
3
votes
1answer
141 views

Improper integral depending of one parameter

I would like to show that $$\int_0^{\infty}\frac{\sin x}{x}e^{-nx} \, dx=\arctan\frac{1}{n}$$ Any help is appreciate!
1
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1answer
111 views

$F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + x^2}dx.$ Show that $\lim\limits_{u\downarrow0}F(u)=f(0)$.

This is an problem from p. 296 of Buck's Advanced Calculus: Let $f$ be continuous on the interval $0\leq x < \infty$ with $|f(x)|\leq M$. Set $$F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + ...
0
votes
1answer
607 views

Dirichlet integral. [duplicate]

I want to prove $\displaystyle\int_0^{\infty} \frac{\sin x}x dx = \frac \pi 2$, and $\displaystyle\int_0^{\infty} \frac{|\sin x|}x dx \to \infty$. And I found in wikipedia, but I don't know, can't ...
2
votes
0answers
72 views

Frullani version of the classic $\log \left( 1 + 2\alpha \cos px + \alpha^2\right)$ integral.

I read in a paper about Frullani integrals the following claim $$ \begin{align*} I & :=\int_0^\infty \frac{1}{x}\log\left(\frac{1 + 2\alpha \cos px + \alpha^2}{1 + 2\alpha \cos qx + ...
9
votes
3answers
385 views

How do I show that $\int_0^\infty \sin(ax) \sin(bx) / x^2 = \pi \min(a,b)/2$

Recently I found a claim saying that $$ \int_0^\infty \left( \frac{\sin ax}{x}\right)\left( \frac{\sin bx}{x}\right) \mathrm{d}x= \pi \min(a,b)/2 $$ from what I can see this seems to be true. I ...
2
votes
5answers
124 views

Evaluate $\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$

Evaluate $$\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$$ by any method. In short I am interested in any method that overcomes the lack of convergence of this integral and gives an "number" to it. ...
3
votes
2answers
329 views

convergence test : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $

I have to check if $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $ is convergent or divergent. My approach was to integrate the function , hence : $\int_{0}^\infty \mathrm ...
4
votes
1answer
83 views

Why does $\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$? [duplicate]

I have found that $$\int\limits_0^1 {\dfrac{{x - 1}}{{\ln x}}} \;\text{d}x=\ln2$$ but I can't prove it. Any hint? Thank you in advance
0
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1answer
86 views

Evaluating $\int_0^{\infty} \log\left(1-2\frac{\cos(2a)}{x^2}+\frac{1}{x^4}\right)^2 \ dx, \quad 0\le a \le \pi $

@Did, what do you recommend me for? $$\int_0^{\infty} \log\left(1-2\frac{\cos(2a)}{x^2}+\frac{1}{x^4}\right)^2 \ dx, \quad 0\le a \le \pi $$ The integration by parts is of no help. What else is ...
3
votes
3answers
137 views

Convergence of $\int\limits_{-\infty}^{\infty}\frac{\sin{x^2}}{x}dx$

I had to prove that the integral $\int\limits_{-\infty}^{\infty}\frac{\sin{(x^2)}}{x}dx$ converges. I thought splitting it to ...
3
votes
1answer
91 views

Prove that if $f_n\to f$ uniformly and $\int_0^\infty|f_n|\le M$, then $\int_0^\infty|f|<\infty$

Again while preparing to calculus I found another interesting question: Prove or give counterexample that if $f_n\to f$ uniformally $[0,\infty]$ and $\forall n\in\mathbb {N}\int_0^\infty|f_n|dx\le ...
3
votes
1answer
73 views

Proving $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$

The question is Prove that for any $n\in\mathbb N$, $\int\limits _0^\infty\frac {\sin^{2n+1}x}{x}dx=\frac 1 {4^n}\binom{2n}{n}\int_0^\infty \frac {\sin{x}}{x}dx$ I don't have any ideas how to ...
0
votes
2answers
170 views

Conditional/Absolute convergence of $\int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$

I need to check conditional/absolute convergence of the integral: $$f(x) = \int_{1}^{\infty}\cos(x^{2})\,\mathrm dx$$ I tried for a long time and I can't understand what I should do. I know that ...
1
vote
2answers
151 views

conditional or absolute convergence of integral

i ran into a few problems where i had to check absolute\conditional convergence of a few integrals. i'm sure theres a method to check this, i just can't find the trick. i wan't help with one of the ...
4
votes
0answers
129 views

Taking an integral of e^(another integral) with respect to the limit of integration of (another integral)

How would I go around integrating $$\int_0^\infty \left( \exp{\left(-\int_{k_0}^{k_0+t} (k_1 + k_2s)(k_3+k_4s)^{s} ds \right) } \right) dt \text{,}$$ where $k_i$ are constants? Is it solvable ...
16
votes
6answers
556 views

Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$

I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that $$ ...
2
votes
1answer
110 views

Laplace transform of sin(at)

Given $f(t)= \sin (at)$ I want to calculate the Laplace transform of $f(t)$. I have determined by using integration by parts twice, that the answer should be $$F(s)= \frac{a}{s^2+a^2}$$ Now I want ...
1
vote
4answers
110 views

divergence of $\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$

i ran into this question and im sitting on it for a long time. why does this integral diverge: $$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$$ thank you very much in advance. yaron.
10
votes
4answers
549 views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
4
votes
1answer
91 views

integral involving modified bessel

I want to integrate this: $$\int_0^{\infty} dt \exp{\left ( a \, t^b\right)} \, I_v {\left ( a \, t^b\right)} $$ where $I_v(.)$ is the modified bessel function of arbitrary order $v$. Can someone ...
0
votes
1answer
69 views

$\int_{0}^{\infty}{\ln(x+3){x^{-2}} \mathrm d x}$ converges?

I ran into this question: show convergence/divergence of: $$\int_{0}^{\infty}{\ln(x+3){x^{-2}} \mathrm d x}$$ I tried for a long time and I'm kind'a lost. according to the answer, it should ...
0
votes
0answers
46 views

Gelfand Shilov vol 1. question: finite part of an integral

how do Gelfand and Shilov prove that the finite part $$ F.P \int_{0}^{\infty}x^{m}dx=0 $$ however i get for non zero lower limit the recurrence of integrals in terms of the Riemann zeta function $$ ...