1
vote
1answer
41 views

Existence of a function with certain integral properties

Does there exist a non-negative Borel-measurable function $g:\mathbb [1,\infty)\to[0,\infty)$ such that \begin{align*} \int_1^{\infty}g(y)^2\,\mathrm dy<&\,\infty,\\ ...
2
votes
3answers
122 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
2
votes
1answer
86 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
5
votes
3answers
105 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
-1
votes
1answer
42 views

Value of line integral [on hold]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
5
votes
5answers
177 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
1
vote
1answer
95 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
9
votes
3answers
147 views

Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$

I have in trouble for evaluating following integral $$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$ It seems really easy, but I ...
3
votes
0answers
29 views

Parameter-dependent integral: Is the following statement true?

Is the following statement true? If so, could anyone provide a reference? Suppose $f(x, \alpha)$ is continuous on $(a, b) \times \{\alpha_0\}$. If there exists $g(x)$ which is continuous on $(a, b)$, ...
3
votes
2answers
116 views

Find $\lambda$ if $\int^{\infty}_0 \frac{\log(1+x^2)}{(1+x^2)}dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}dx$

Problem : If $\displaystyle\int^\infty_0 \frac{\log(1+x^2)}{(1+x^2)}\,dx = \lambda \int^1_0 \frac{\log(1+x)}{(1+x^2)}\,dx$ then find the value of $\lambda$. I am not getting any clue how to proceed ...
1
vote
3answers
29 views

Growth restriction for nonnegative, continuous functions whose integrals on $\mathbb{R}$ are bounded

When we have a nonnegative, continuous function $f(x)$ whose integral over all real numbers $\mathbb{R}$ is bounded, like: $$\int_{-\infty}^{\infty}f(x)dx = A< \infty $$ with $A \in \mathbb{R}$ ...
1
vote
0answers
52 views

Question on integral

I need a confirmation and answer about the following problem: If we have $ g(x)=\ln x+{ x }^{ -1/2 }{ 1 }_{ x\le 1 } $ I'm trying to determine $ \int _{ 0 }^{ +\infty }{ \ln x+{ x }^{ -1/2 }{ 1 }_{ ...
0
votes
1answer
27 views

How to identify continuity or discontinuity of an [Definite] integral?

How can I figure out whether an improper integral converges based on the discontinuities in the integrand? For instance, these two both have discontinuities within the intervals of integration, and ...
1
vote
0answers
152 views

$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$

The following integral bothers me since weeks: $$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$ Has any body a suggestion for this integral. $Re\; s >0$ sufficiently large and $s'$ an ...
8
votes
1answer
99 views

Interesting sum-integral equality

Is there an elementary proof of $$\lim_{n \to \infty} \int_0^\infty e^{-\alpha x^2} \frac{\sin((2n + 1)x)}{\sin x} dx = \pi\left(\frac{1}{2} + \sum_{k = 1}^\infty e^{-\alpha k^2 \pi^2}\right),$$ where ...
1
vote
0answers
48 views

Choose appropriate contour for a complex integral

I have a problem to solve integral $$ I = \int^{\infty}_0 \frac{\mathrm{d}x}{(x-z)(1+x^2)^{\kappa+2}} $$ I can solve the same integral with borders $-\infty$ to $\infty$ using residue theorem but ...
8
votes
1answer
256 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
1
vote
1answer
59 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
2
votes
1answer
26 views

fourier transform of scaled function

let us consider following example one thing which i did not understand is where absolute value of $a$ came from?ok if we have $\int^{\infty}_{-\infty} x(a*t)*e^{-j\omega*t}dt$ then we may have ...
2
votes
0answers
53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
1
vote
2answers
74 views

Compute $\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$ as a function of $\Gamma(n)$

Is it possible to compute this integral $$\int_{0}^{\infty}e^{-tz}(z+d)^{n-1}dz$$ as a function of complete gamma $\Gamma(n)$. If possible, I'm looking for a closed form solution. Thanks!
-2
votes
0answers
56 views

Convolution Integral

Does someone know how to solve this convolution integral? $$V(x)=c_1\int\limits_{-\infty}^\infty \left(r(\tilde x)+\dfrac{c_2}{n(\tilde x)}-n(\tilde x)\right)\left(\sqrt{c_3+(x-\tilde x)^2}-|x-\tilde ...
0
votes
1answer
50 views

Evaluating integral involving Bessel function.

Evaluate $$\int_0^{\infty } \frac{2^{\frac{r}{\delta }} \left(2^{\frac{r}{\delta }}-1\right) r\ e^{-\frac{\alpha ^2+\left(2^{\frac{r}{\delta }}-1\right)^2}{2 \beta ^2}}\log 2 }{\beta ^2 \delta } ...
2
votes
1answer
74 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
1
vote
2answers
88 views

Limit of a function with a defined integral 3

Let $F$ be function defined as an integral $$F(x)=\int_{1}^{\infty}\dfrac{t^ke^{-xt}}{1+t^{5}}\textrm{d}t \quad \forall k\in \mathbb{N},\ x>0$$ Show that $\lim_{x\to \infty}F(x)=0$ ...
2
votes
2answers
110 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
0
votes
0answers
29 views

Gaussian integral involving $\cos\circ\sin$

I stumbled upon an integral of the form $$\int_{\mathbb R} e^{-x^2/2}\cos(a\sin (bx+ic))\,{\mathrm d}x$$ for some real constant $a,b,c$. Has anybody ever seen such an integral? Mathematica doesn't ...
2
votes
3answers
101 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
3
votes
1answer
162 views

How to find $\int_0^{\pi}\frac{\sin n\theta}{\cos\theta-\cos\alpha}d\theta$

I was doing some work in physics and I came up with a definite integral. I tried everything I could but couldn't solve the integral. The integral is $$ \int_0^\pi {\sin\left(n\theta\right)\over ...
4
votes
3answers
262 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
5
votes
3answers
172 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
1
vote
2answers
82 views

Prove that $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$ [closed]

Consider the next function: $$\Gamma\left(t\right)=\int_{0}^{+\infty}x^{t-1}e^{-x}dx.$$ Prove that $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$
1
vote
1answer
27 views

Does the integral in the formal 2D Fourier transform of the logarithm converge?

If $k$ is a nonzero vector in $\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance.
2
votes
2answers
53 views

Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^{\infty}\frac{e^{itx}}{\pi(1+x^2)}dx$ using residue theorem. Is there any other way to calculate this integral (for someone who ...
2
votes
0answers
84 views

Calculate the Gauss integral without first squaring it

We know that the integral $$I = \int_{-\infty}^{\infty} \mathrm{d}x e^{-x^2}$$ can be calculated by first squaring it and then treat it as a 2-d integral in the plane and integrate it in polar ...
2
votes
4answers
136 views

Integrate $\tan^a(u)$ from 0 to $\pi/2$

I have this problem: $$\int_{0}^{\infty} \frac{x^a}{x^2+1}dx$$ with $0<a<1$. I get the integral $$\int_{0}^{\frac{\pi}{2}}\tan^a(u)du,$$ But I can't solve any of the two problems, how can I ...
0
votes
0answers
41 views

proving improper integral converge

I'm trying to prove the following integral converge: $$ \int_{0}^{\infty}\frac{e^{-\frac{1}{x}}-1}{x^\frac{2}{3}} $$ since 0 and $\infty$ are the problematic points I've done this: $$ ...
4
votes
3answers
524 views

Prove the equation

Prove that $$\int_0^{\infty}\exp\left(-\left(x^2+\dfrac{a^2}{x^2}\right)\right)\text{d}x=\frac{e^{-2a}\sqrt{\pi}}{2}$$ Assume that the equation is true for $a=0.$
1
vote
1answer
28 views

Explain what the teacher did, convergence of improper integral

I'd like someone to explain what the teacher did, because I'm not sure I understand. Basically, the question is for which values of $p$, does the integral $$\int_{1}^{\infty} \frac{dt}{t \log ...
0
votes
0answers
11 views

Numerically integrate an improper integral involving product of two bessel functions and a singularity

I am working on an elasticity problem which requires solving an integration with a rather complex kernel involving production of two Bessel's functions of the first kind (zeroth order) and a ...
2
votes
1answer
45 views

if $\int _a^{\infty }\left|f\left(x\right)\right|\:dx$ converges then $\int _a^{\infty }f\left(x\right)\:dx$ also converge.?

if $\int _a^{\infty }\left|f\left(x\right)\right|\:dx$ converges then $\int _a^{\infty }f\left(x\right)\:dx$ also converge. I wonder what i missing. we proved in our course thar for every $g$ >$f$>0 ...
6
votes
3answers
480 views

How do you integrate the reciprocal of square root of cosine?

I encountered this integral in physics and got stuck. $$\int_{0}^{\Large\frac{\pi}{2}} \dfrac{d\theta}{\sqrt{\cos \theta}}.$$
0
votes
0answers
22 views

determine the nature of improper intgral

I would like to determine the nature of $B$ without calculating it . $$ B= \int_0^1 |1-t^{a}|^{b} dt . $$ using Taylor's Expansion in $0$ two times (for $a$ n $b$ ) would give us : ...
1
vote
1answer
38 views

Convergence of improper integral with logarithm

I would like to determine the nature of $A$ without calculating it. $$ A= \int_0^1 \ln(1-t^{a}) dt . $$ In $t=1$ we have a problem, so how should I proceed?
3
votes
3answers
132 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
2
votes
1answer
54 views

Integration of exponential functions: $\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$

How I am to solve this integral? I am not able to use any of the methods. $$\int_0^\infty e^{-({x^2}/{y^2})-y^2}\; dx$$
2
votes
4answers
83 views

Definite Integral: $\int_0^1\frac{x^{2} -1}{\log x}\,dx$ [duplicate]

How to figure out the following integral? I have not been able to solve it from some time. $$\int_0^1\frac{x^{2} -1}{\log x}\,dx$$
4
votes
4answers
85 views

$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?

Question I'm trying to solve to prepare for an exam. I need to find out if $\displaystyle\iint_{\mathbb R^2} \frac{dx\,dy}{1+x^{10}y^{10}}$ diverges or converges. What I did: I switched to polar ...
3
votes
1answer
66 views

Check my proof $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2) \, dx \, dy$ diverges

I am trying to prove that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \sin(x^2+y^2)\, dx\,dy$ diverges and I did it like this: $x=r\cos \theta$, $y=r\sin\theta$, $\theta \in [0,2\pi]$, $r\in ...
2
votes
2answers
80 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 ...