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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2
votes
3answers
642 views

Does $f(x)\in L^1$ imply that $\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0$?

Suppose that $f(x)$ is $L^1$ and R- integrable function, problem is to resolve if it is possible existence of such a $f(x)$ that: $$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ...
15
votes
3answers
361 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
2
votes
2answers
100 views

Integral from $0$ to $\infty$ of $\ln(x)/e^x$

Show $$\int_0^\infty \frac{\ln(x)}{e^x} = -\gamma$$ (gamma is Euler-Mascheroni constant). Can anyone please prove this result? Also $$ \int_0^\infty \frac{\left( \ln(x) \right)^2}{e^x}\mathrm dx. ...
3
votes
2answers
294 views

Improper Integral of $x^2/\cosh(x)$

I need to compute the improper integral $$ \int_{-\infty}^{\infty}{\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x} $$ using contour integration and possibly principal values. Trying to approach this as ...
0
votes
1answer
214 views

Clarification of Cauchy Principal Value and use of Contour Integration

I am evaluating the improper integral $\int_{-\infty}^\infty{\frac{\sin^3 x}{x^3}dx}$; I am also told to show that this is equal to its principal value, and use this fact to evaluate the integral. I ...
2
votes
2answers
44 views

Is my divergence test correct?

This idea came to me while looking at the following graph of $f=\frac{1}{x}$: Now, the definite integral of $f$ from $1$ to $n$ is smaller than $f(1)+f(2)...f(n)$, from the graph above. But since ...
0
votes
1answer
38 views

An Imporper Integral

I am to find out whether the following Improper Integral converges: $$\int_2^\infty \frac{e^{x/4}}{x^3{ln}^5x}\,dx\quad$$ Things that I've tried: Comparison with $$\frac{1}{x^3{ln}^5x}$$ Or ...
0
votes
1answer
36 views

Equality of two trigonometric integrals on [0,1]

I need to show, that : $$\int_0^1 \cos(x^2)~\mathrm{d}x = \frac{1}{2} \int_0^1 \frac{\cos x}{\sqrt x}~\mathrm{d}x$$ But frankly I cannot see way to solve it. The right-side integral is ...
0
votes
0answers
61 views

Integration by Residue Theorem - Is This Integral equal to Zero?

$$ \int_{-\infty}^{+\infty}\frac{\sin(x)}{2x^{5}-3jx^{3}+2x}dx=0 $$ ...
4
votes
1answer
63 views

Convergence of $f(t)= \frac{\exp(it)}{t^a}$

Let $$f(t)= \frac{\exp(it)}{t^a}.$$ For what values ​​of $a$ does the integral $\displaystyle\int_{0}^{+\infty}f(t)dt$ converge? For $a>0$ it's clear with an integration by parts ...
12
votes
2answers
162 views

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

I need to evaluate the following integral with a high precision: $$ I=\int_{0}^{\infty}\left[% {\pi^{2} \over 6} - {\rm Li}_2\left({\rm e}^{-x}\right) -{\rm Li}_2\left({\rm ...
1
vote
2answers
111 views

Does $\int_{1}^{\infty}|\sin(x)/x|dx$ converge?

$$\int_{1}^{\infty} \left| \frac{\sin(x)}{x}\right|dx$$ I'd like to know if this integral converges or not. I tried Wolframalpha but it didn't give me answer.
0
votes
3answers
123 views

Calculate the limit of integral

I'm doing exercises in Real Analysis of Folland and got stuck on this problem. I don't know how to calculate limit with the variable on the upper bound of the integral. Hope some one can help me solve ...
1
vote
3answers
112 views

Evaluating improper integrals of odd functions with hyperbolic and circular elements

The integral $$ \int_{0}^{\infty} {\sin\left(\omega t\right) \over \cosh^{2}\left(t/\sqrt{2\,}\,\right)}\,{\rm d}t $$ with $ \omega >0 $ is an odd function in variable t. This precludes any ...
0
votes
3answers
131 views

Gaussian-Like integral

What is the integral of this? $$\int_0^\infty xe^{-(ax^2+bx)}\,\mathrm{d}x$$ $a$ and $b$ are positive integers.
0
votes
1answer
32 views

Show $\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}$?

is there a way to show the inequality \begin{equation}\int_{M}^{\infty}\left|\frac{e^{-x^a}\cos(Cx)}{x^{b+1}}\right|dx<\frac{1}{aM^{a+b}e^{M^a}}\end{equation} for positive constants $M$ and $C$ ...
3
votes
0answers
338 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
1
vote
1answer
283 views

How to derive integrals with error function?

How to derive this integral $\int_{-\infty}^{\infty}erf(\lambda x)\mathcal{N}(\mu, \sigma ^2)dx$ and this $\int_{-\infty}^{\infty}(erf(\lambda x)-const)^2\mathcal{N}(\mu, \sigma ^2)dx$ where ...
2
votes
1answer
144 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
0
votes
2answers
52 views

Mean residual life: (improper) integration by parts involving p.d.f

Let a random variable $X$ be distributed with p.d.f $f(\cdot)$. I want to derive the following equality for the mean residual life: $$m(x)\equiv\frac{\int_{x}^{\infty}(z-x) ...
2
votes
1answer
71 views

Integral with the incomplete upper gamma function

Can anyone help me integrate this? $$\int_0^1 \frac{1}{x^{1/p}} \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{m/n-1} \Gamma\;\left(A, \left[\frac{1-x^{1/p}}{x^{1/p}} \right]^{1/n}\right) ...
3
votes
0answers
102 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
5
votes
3answers
253 views

How to compute $I_n=\int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}$?

I'd like to compute: $$ I_n = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{x^{2n}}{\cosh^2 x}. $$ We have, quite easily: $$ I_0 = \int_{-\infty}^{+\infty}\mathrm{d}x\frac{1}{\cosh^2 x}=\left[\tanh ...
4
votes
1answer
117 views

what is your favorite first-year calculus improper integral?

I am writing a take home quiz for a first-year university level calculus course. This quiz deals with improper integrals. I would like to know if you would be kind enough to share your favorite ...
5
votes
3answers
228 views

Proving $\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = \frac{\Gamma(1-r)}{r}(1-a^2)^{r/2} \cos(r \arctan(a))$

does anyone have an idea or a guess how to prove the following equation: $$\int\limits_{0}^{\infty}\frac{1-\text{e}^{-x}\cos(ax)}{x^{r+1}}\operatorname d\!x = ...
2
votes
0answers
150 views

on the convergence of a certain integral

If I have an entire function $\phi$ such that it is of exponential order zero. I.e for all $\rho > 0$ we get $|\phi(s)|\le C_\rho e^{|s|^{\rho}}$. Furthermore, I have an extreme decay in the Taylor ...
0
votes
2answers
767 views

integral of 1/(sqrt(e^x)) from 0 to infinity(Improper integral)

I'm taking Calculus 1 course and I'm having problems with the following integrals(Improper integrals) $\displaystyle\int_0^{\infty} \frac1{\sqrt{e^x}}$ dx $\displaystyle\int_0^1 e^\frac1x$ dx ...
0
votes
1answer
41 views

range of a function defined by expectation

Given $c>0$ and define a function $f:(0,\infty)\mapsto \mathbb{R}$ by \begin{equation} f(\sigma)=\frac1{\sqrt {2\pi}\sigma}\int_{-\infty}^\infty\max\{-c,\min\{x,c\}\}^2e^{-\frac{x^2}{2\sigma^2}}. ...
0
votes
3answers
39 views

Computing the limit of this function

So I have an improper integral: $$ \int_0^\infty \frac{13x}{x^2+1}-\frac{65}{5x+1} dx $$ I have solved the integral into this: $$ \lim_{t \to \infty} ...
1
vote
0answers
36 views

How to find $\int_0^\infty \frac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
4
votes
4answers
145 views

How find this integral $F(y)=\int_{-\infty}^{\infty}\frac{dx}{(1+x^2)(1+(x+y)^2)}$

Find this integral $$F(y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x+y)^2)}$$ my try: since $$F(-y)=\int_{-\infty}^{\infty}\dfrac{dx}{(1+x^2)(1+(x-y)^2)}$$ let $x=-u$,then ...
2
votes
2answers
110 views

How to prove that $\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1$?

I have some trouble to prove that $$\int_{-1}^{1}\exp\left(\frac{1}{x^2-1}\right) \ dx=1\ ? $$
0
votes
4answers
135 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
2
votes
1answer
65 views

improper integral $\int_{0}^{\infty}\frac{2x}{e^{x}-e^{-x}}dx$

I have a problem with my solution. $\int_{0}^{\infty}\frac{2x}{e^{x}-e^{-x}}dx=\int_{0}^{a}\frac{2x}{e^{x}-e^{-x}}dx+\int_{a}^{\infty}\frac{2x}{e^{x}-e^{-x}}dx$. So in first integral, if I compare it ...
1
vote
2answers
99 views

Integral of $x^2e^{-ax^2}$

Hey guys I need to find the following integral using integration by parts and not the gamma function. Also there is an a constant a in the exponential function. So it is actually $x^2e^{-ax^2}$. ...
3
votes
5answers
222 views

How to evaluate this Trig integral?

I need to find the definite integral of $\int(1+x^2)^{-4}~dx$ from $0$ to $\infty$ . I rewrite this as $\dfrac{1}{(1+x^2)^4}$ . The $\dfrac{1}{1+x^2}$ part, from $0$ to $\infty$ , seems easy ...
3
votes
2answers
81 views

Proving an inequality on $ \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$

EDIT : I have posted a proof below that needs to be reviewed. Some definitions Let $$\begin{array}{ccccc} f & : & \mathbb R_+^* & \to & \mathbb R_+^* \\ & & x & \mapsto ...
0
votes
1answer
55 views

How can I find this integral

How can I find this integral? please help $$I=\int_0^1\frac{e^{-\sqrt{x}}}{\sqrt{x}}\ dx.$$
1
vote
1answer
63 views

integral with gaussian function

I am trying to evaluate the following integral: $$ \int_0^\infty{z^{m-1}\over\left[1+\left(\eta z\right)^n\right]^p}e^{-(z-b)^2\over c}\,{\rm d}z, $$ where the integration is w.r.t. to $z$, and the ...
2
votes
0answers
127 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
2
votes
1answer
68 views

Improper integral $\int^{1}_{0} \frac{x}{\sin{(x^{p})}} dx$

I have an improper integral with $p > 0$, $$\int^{1}_{0} \frac{x}{\sin{(x^{p})}} \ dx$$ and I want to find for which $p$ the integral exists. Now we should consider when $p = 1$ and when $p \not= ...
2
votes
1answer
102 views

Prove that $\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n!\zeta (n+1)$

I have encountered the following identity on Wolfram alpha and I fail proving it (with $n \in \mathbb N^*$) $$\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n! \zeta(n+1)$$ I tried to rewrite the ...
1
vote
1answer
51 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
2
votes
2answers
130 views

Which my step is incorrect to prove $\int_{-\infty}^\infty xe^{-x^2}dx = 0$?

To prove $\int_{-\infty}^\infty xe^{-x^2}dx = 0$, my solution is Let $y=x^2$, then $dy=2xdx$. $\begin{align} \int_{-\infty}^\infty xe^{-x^2}dx &= \frac{1}{2}\int_{-\infty}^\infty e^{-x^2}2xdx ...
11
votes
1answer
390 views

integral $\int_{0}^{\infty}\frac{\cos(\pi x^{2})}{1+2\cosh(\frac{2\pi}{\sqrt{3}}x)}dx=\frac{\sqrt{2}-\sqrt{6}+2}{8}$

Here is a seemingly challenging integral some may try their hand at. $$ \int_{0}^{\infty} {\cos\left(\pi x^{2}\right)\over 1 + 2\cosh\left(\,2\,\pi\,x\,/\,\sqrt{\,3\,}\,\right)}\,{\rm d}x ...
20
votes
5answers
3k views

How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/a))\,dx$?

How do I find $$\large\int_{0}^{\infty}e^{-\left(ax+\frac{b}{x}\right)}dx$$ where $a$ and $b$ are positive numbers? This is not a homework question. I will be quite happy if somebody can come up ...
4
votes
2answers
313 views

$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?

Let $f: \mathbb R \rightarrow \mathbb R $ be a continuous function such that $\int_{0}^{\infty} \,f(x) dx$ exists. Then Prove that incase (i) $f$ is a non negative function, then ...
1
vote
3answers
82 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
0
votes
1answer
109 views

how to integrate $\int \frac{\sin x}{x}$ in $[0,1]$ [closed]

I would appreciate if somebody could help me with this problem $$\int_{0}^{1}\frac{\sin{x}}{x}dx$$ here using Taylor series I got $\sum_{0}^{\infty} $$\frac{(-1)^n}{(2n+1)!(2n+1)} $ then what to do? ...
2
votes
1answer
150 views

Integration limits of the double integral after conversion to the polar coordinates

I want to solve the following double integral: $$\int_0^{\infty}dx\int_{-\infty}^{\infty}dy\,f(x,y).$$ And for example I made a conversion to the polar coordinates, $x=r\cos{\theta}$ and ...