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
90 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 $\lim_{x\to \infty}F(x)=0$ ...
1
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0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
2
votes
2answers
111 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
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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 ...
3
votes
3answers
124 views

How to calculate the integral $\int_{-1}^{1}\frac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

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
170 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
272 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
7
votes
3answers
205 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
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1answer
41 views

Evaluating improper integral expression

Can anybody please guide me in evaluating this expression, for my research work. I have tried a lot but in vain. Both the integrals involve gamma functions and even wolfram said that the time limit ...
0
votes
2answers
38 views

Evaluating Improper integral

Using the equation $\frac{1}{\sqrt{x}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\exp(-\alpha^{2}x) \, \mathrm{d}\alpha$, for $\alpha>0$, Compute the two integrals $$\int_{0}^{\infty} ...
1
vote
2answers
85 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 $\Gamma\left(t+1\right)=t\ \Gamma\left(t\right)\quad\forall{t>0}.$
0
votes
1answer
35 views

Convergence of improper integral depending on parameter [closed]

For what $a>0$ $(\frac{1}{\sin x})^a$ is integrable on $(0,\frac{\pi}{2})$?
2
votes
1answer
104 views

Proving $\displaystyle \int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$ Exists

As the title says, I need to show that $$\int_0^{1} \sin\left(x + \frac{1}{x}\right)\, dx$$ exists. After performing the substitution $x = 1/u, dx = -1/u^2 du$, the integral becomes ...
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
60 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 ...
0
votes
2answers
44 views

Converging improper integral have sequence with limit of zero

I've the following statement: Let $ f : \mathbb{R} \rightarrow \mathbb{R} $ integrable function. If $ \int_0^\infty f(t)dt $ converge, does sequence $ (x_n)\in \mathbb{R} $ exist such that: $ ...
2
votes
0answers
90 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
138 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 ...
2
votes
0answers
35 views

Help! Improper integral convergence (values of P)

I'm quite lost on the following problem: $$\int_{0}^{\pi/2} \frac{sin^2(x)}{x^{p^2-3p-7}}dx$$ I can't figure out how to work out the given answer. Please help me!
0
votes
0answers
42 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
526 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
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1answer
29 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
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0answers
17 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 ...
2
votes
0answers
27 views

Improper integral $\int^{\pi}_{0}\frac{d\theta}{|\cos(\tau - \theta)|^s}<\infty $

For some fixed $s<1$, how can we proof $$\sup\{\int^{\pi}_{0}\frac{d\theta}{|\cos(\tau - \theta)|^s} : \tau\in[0,\;2\pi)\;\}<\infty$$
12
votes
3answers
315 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
2
votes
1answer
23 views

Using Taylor's series in imporper integrals

Is it possible to simplify an improper integral using Taylor's series? How can I prove this procedure is correct? For example, take $$f(\alpha)=\int_0^{\infty} ...
6
votes
3answers
494 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
17 views

Continuity of a function defined by an improper integral.

What is the result that allows us to say the following: In order to show that some function $f(x)= \int_0^{+\infty}g(x,t)dt$ is continuous on $[0,+\infty)$ we show that $g$ is continuous on ...
2
votes
0answers
23 views

Improper integral when the integrand goes to infinity.

Is it true that if $$\lim_{x\to +\infty} f(x)=+\infty$$ then $f $ can not be integrable at the neighborhood of $+\infty$, hence the improper integral $\int_0^{+\infty}{f(x)dx}$ does not exist?
0
votes
1answer
34 views

How to calculate the value of $\int_{-\infty}^{\infty} y(t)dt$?

For a function $g(t)$, $\int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2}$ for any real value $\omega$. If $y(t)=\int_{-\infty}^{t}g(\tau)d\tau$ then how to calculate the value of ...
1
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0answers
24 views

Existence of measurable fuction on non-atomic measure space whose integral is infinity

Let $(X,M,\mu)$ be non atomic measure space with $\mu(X)>0.$ Show that there is a measurable function $f:X\to [0,\infty),$ for which $\int f(x)d\mu(x)=\infty.$ No idea at all. I am preparing for ...
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
4answers
159 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.
1
vote
1answer
25 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
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
88 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
70 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
81 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 ...
0
votes
1answer
16 views

Showing the convergence of improper integral.

Hello I have to show that this improper integral is convergent: $$\int_{0}^{1} \frac{e^{\frac{-1}{x}}}{x^2} dx$$ , but I don't have any starting ideea. Any tips would be great, thank you.
4
votes
3answers
68 views

How to show that $\int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt = \pi^{1/2} \frac{ \Gamma(s-1/2)}{\Gamma(s)}$

I want to compute the integral $$ \int \limits_{-\infty}^{\infty} (t^2+1)^{-s} dt $$ for $s \in \mathbb{C}$ such that the integral converges ($\mathrm{Re}(s) > 1/2$ I think) in terms of the Gamma ...
1
vote
3answers
41 views

Find convergence of improper integral.

Hello I have to find the convergence of this improper integral: $$\int_{e}^{\infty} \frac{1}{x\log^2x} dx$$ So I started by doing the following: $\lim \limits_{x \to A} \int_{e}^{A} ...
3
votes
1answer
98 views

Asymptotic behavior of a sequence of integrals

I am interested in the asymptotic behavior of sequences $(I_n)$ and $(J_n)$ as $n \rightarrow \infty$, where $$I_n = \int_{1}^{\infty}\frac{e^{-nx^2}}{x^2}\, dx,$$ and $$J_n = ...
1
vote
2answers
24 views

Is there a smooth function with an asymptote at zero and integrable over $]0,\infty[$?

If you look at functions of the form $1/x^k$, $k>0$, it seems you can't have your cake and eat it too. If the integral of $1/x^k$ converges on $[1,\infty[$, then it diverges on $]0,1]$ and ...
2
votes
1answer
114 views

Sophomore's dream: $\displaystyle\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$

In the solution of the so-called sophomore's dream, one of the key steps is to compute $$\int_0^1 x^n (\log x)^n$$ using the change of variables $x = \exp\left(-\frac{u}{n+1}\right)$ to obtain the ...
0
votes
2answers
54 views

Finding $(p,q)$ such that $\frac{x^p}{1+x^q}$ is integrable on $(0,+\infty)$

I'm trying to show that $f(x) = \frac{x^p}{1+x^q}$ is integrable on $(0,\infty)$ if and only if $p > -1$ and $q-p > 1$. So on $[1,\infty)$ we can compare with $g(x) = x^{p-q}$ which is ...
6
votes
3answers
112 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
1
vote
1answer
30 views

Which singularities are integrable

Consider a function $f: \mathbb{R}^n \to \mathbb{R}$ that has a singularity at point $x_0$ such that $\lim_{x\to x_0} f(x) = +\infty$. When we can be sure that this singularity is integrable? (you can ...
1
vote
1answer
37 views

Regularizing conditionally convergent integrals

For functions $f:\mathbb{R}\to \mathbb{R}$ the definition of a convergent improper integral is straightforward: the integral $\int_\mathbb{R} f(x)dx$ converges iff $$\lim_{(a,b)\to (-\infty, ...