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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3answers
54 views

Calculus $\int_0^{+\infty}\frac{\sin^2x}x\mathrm dx$ [on hold]

Calculus $$\int_0^{+\infty}\frac{\sin^2x}x\mathrm dx$$ I have just approached to improper integrals, and it may be rather complex to me.
1
vote
0answers
31 views

Integrals with error function and exponentials

I'm trying to solve the integrals below: $$\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{x}{\sqrt{x^2+y^2}}\cdot \operatorname{erf}\left(m\cdot\sqrt{x^2+y^2}\right) \cdot \exp(-a\cdot ...
0
votes
0answers
37 views

How can I find these integral value of $y(t) = 1 - e^{-2t}sin(4t)$?

How can I find... the integral square value the integral absolute value $y(t) = 1 - e^{-2t}sin(4t)$ ? Please help. Thank you
0
votes
0answers
14 views

Principal value with truncation in $y$-direction

The Cauchy principle value uses truncation in $x$-direction, e.g$$PV\int_{-1}^1 \frac1x \, \mathrm{d}x = \lim_{\varepsilon \searrow 0} \int_{-1}^{-\varepsilon} \frac1x \, \mathrm{d}x + ...
-1
votes
0answers
11 views

Intution for Dirichlet test for improper integrals

Given Theorem states that If $\phi$ is bounded and monotonic in $[a,\infty)$ and tends to 0 as x tends to $\infty$ and $\int_a^Xfdx$ is bounded for $X\geq$ a, then $\int_a^{\infty}f \phi dx$ is ...
2
votes
3answers
256 views

One Step Forward from Gaussian Integral

Now to solve the integral $ \int_0^\infty e^{-x^2} \, dx $ has become a simple task for us. But how can we solve this integral: $$\int_0^\infty e^{-x^3} \, dx $$
5
votes
5answers
189 views

Integrating $ \int_0^\infty \frac{x^5}{e^x+1} \, dx $

This improper integral has stumped me and not many integration problems give me problems. However, this one made me think over my limit. Finally, as I could not even get started on this problem, I ...
0
votes
1answer
47 views

Evaluating the integral $\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt$

It is stated that (for $\lambda>0$) $$\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{2i\mu t}\frac{{\sin^2{\lambda t}}}{\lambda t^2}}dt = 1-\frac{|\mu|}{\lambda}$$ for $ 0\leq|\mu|\leq\lambda$, and zero ...
4
votes
2answers
85 views

Evaluate the integral $\int_0^\infty x^{t-1}e^{-\beta x}dx$

I want to evaluate the following integral $$\int_0^\infty x^{t-1}e^{-\beta x}dx$$ where $\beta$ is a complex number. Now, if $\beta$ was real, we could just set $y = \beta x$ and we will reduce to ...
1
vote
3answers
71 views

How do I evaluate this improper integral $\int_{-1}^{1}\frac{dx}{(2-x)\sqrt{1-x^{2}}}$

Given integral is $$\int_{-1}^{1}\frac{dx}{(2-x)\sqrt{1-x^{2}}}.$$ I tried to split it up at $0$, but I donot know what to do ahead. Thanks.
3
votes
2answers
53 views

For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, ...
25
votes
4answers
518 views

Intuition behind an integral identity

A proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...
2
votes
4answers
88 views

Convergence of improper integral $\int_{0}^{\frac{\pi}{6}}\dfrac{x}{\sqrt{1-2\sin x}}dx$

I'm trying to determine whether the following improper integral is convergent or divergent. $$ \int_{0}^{\pi/6}\frac{x}{\sqrt{1-2\sin x}}dx $$ At first, I substituted $t=\dfrac{\pi}{2} - x $ and ...
0
votes
0answers
34 views

Find the Fourier transform of the given memory function in the limit volume $V\rightarrow\infty$

The memory function is given by, \begin{equation} \mu (t)=(8\pi e^{2}/3V)\sum_{\vec{k}}|f_{\vec{k}}|^{2}\cos (ckt) \end{equation} where $V$ is the volume, $f_{\vec{k}}$ is the form factor. In this ...
-1
votes
1answer
115 views

$ \int_{-\infty}^{+\infty}e^{-a^2t^2-b^2/t^{2}}\mathrm{d}t=\frac{\sqrt{\pi}}a\:e^{-2ab}$ [closed]

I want to show that $$ \int_{-\infty}^{+\infty}e^{-a^2t^2-b^2/t^{2}}\mathrm{d}t=\frac{\sqrt{\pi}}a\:e^{-2ab}. $$
3
votes
2answers
108 views

Integrating $ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$

I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?: $$ \frac{{ \int_{0}^{\infty} e^{-x^2}\, ...
2
votes
0answers
15 views

Closed form for an improper integral

Can one deduce a general form for the integral: $$\mathcal{J}(a)=\int_0^\infty \frac{\sin (x^2+ax)}{x}\, {\rm d}x$$ I guess that the Feymann trick should do here. Differentiating $J$ with respect to ...
0
votes
1answer
14 views

integration involving Hermit function

Iam trying to evaluate the integration of the following two products .. every one by itself where Hn-1 is the Hermite function . All my tries ends with a zero value for the two integrations, but it ...
4
votes
4answers
84 views

To compute improper integral $\int_3^{5}\frac{x^{2}\, dx}{\sqrt{x-3}{\sqrt{5-x}}}$

I am given improper integral as $$\int_3^{5}\frac{x^{2}}{\sqrt{x-3}{\sqrt{5-x}}}dx$$ DOUBT I see that problem is at both the end points, so i need to split up the integral. But problem seems to me ...
3
votes
1answer
26 views

How do I compute the improper integral $\int_0^{1/e}\frac{dx}{x(\log x)^{2}}$

The given integral is $\int_0^{1/e}\frac{dx}{x(\log x)^{2}}$ ATTEMPT I see that problem is at $0$, so I write the integral as $$\lim_{t \to 0^{+}}\int_{0+t}^{1/e}\frac{dx}{x(\log x)^{2}}$$ Now I ...
4
votes
2answers
93 views

Calculating $\int_0^{\infty} \frac{\log^2(1 - e^{-x})\:x^5}{e^x - 1} \: dx $ [duplicate]

I am having trouble calculating the following improper integral: $$\displaystyle \int\limits_0^{\infty} \frac{\log^2(1 - e^{-x})x^5}{e^x - 1} \, dx $$ Can someone give me a way that I can calculate ...
3
votes
1answer
44 views

Convergence of the integral $\int_0^1 \frac {1}{x\sqrt {1+x^\beta}}dx$

Is my integral-convergence contradiction proof valid? I have to brush up on my proof making. I am a little rusty. I was not sure if the following really held up. I wanted to prove the following is ...
5
votes
1answer
56 views

Show $\int_{0}^{1} \frac{\ln x}{1-x}dx$=$\sum_{1}^{\infty}\frac{1}{n^2}$ and converges

I found this question a) show that the follow integral converges: $\int_{0}^{1} \frac{\ln x}{1-x}dx $ b) $\int_{0}^{1} \frac{\ln x}{1-x}dx$=$\sum_{1}^{\infty}\frac{1}{n^2}$ for the first ...
0
votes
2answers
25 views

To check whether improper integral converges or not $\int_{-1}^1 \frac{(x-1)}{x^{5/3}}dx$

Given integral is $$\int_{-1}^1 \frac{(x-1)}{x^{5/3}}dx$$ ATTEMPT Since there are no problem spots here. so i evaluated integral directly and got some answer. But textbook says integral is ...
0
votes
1answer
28 views

Examine convergence of improper integral

How do i test out convergence of improper integral Given integral is $$\int_0^{\infty}\frac{x^m\cos(ax)}{(1+x^n)}dx$$ Answer is the given improper integral is convergent if $-1 < m < n $. I ...
1
vote
2answers
32 views

bound of integrable function

I want to prove the following conjecture: if an integrable function $f(x)$ is continuous on (0,T] and unbounded at $x=0$, then there exists positive $M$ and $\alpha\in(0,1]$ such that $$ |f(x)|\leq ...
2
votes
3answers
64 views

Improper Integration of A Non-even Non-odd Function From $0$ to $\infty$

I am trying to calculate the integral: $$\int_{0}^\infty \frac{x^2dx}{1+x^7}$$ I used to face this type of integration with even integrand, but the function here is not even nor odd! Is there a trick ...
1
vote
5answers
54 views

does $\int_0^\infty \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx$ converge

does the following integral converges? $\int_0^\infty \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx$ I calculated $$\int \frac{x-\arctan(x)}{x(1+x^2)\arctan(x)} \,dx = \ln(\arctan(x)) - \ln(x) + ...
0
votes
1answer
62 views

Trying to find $\int_{0}^{\infty} \frac{c y^2}{1+c y^2}\frac{1}{\sqrt{2 \pi}} e^{-\frac{(y+c)^2}{2}} dy$

I am trying to find the following integral \begin{align} \int_{0}^{\infty} \frac{c y^2}{1+c y^2}\frac{1}{\sqrt{2 \pi}} e^{-\frac{(y+c)^2}{2}} dy \end{align} where $c>0$. I was able to find the ...
1
vote
1answer
53 views

Change of variable leads to contradiction for an elementary integral.

Integral $I_1(\alpha,\beta)=\int_0^\infty t^\alpha \exp(-i t^{\beta}) ~dt$ converges for $-1<\alpha<\beta-1$. By introducting $u = t^\beta$ the integral is reduced to ...
6
votes
4answers
147 views

Solving $\lim_{n\to\infty}(n\int_0^{\pi/4}(\tan x)^ndx)$?

$$f(x)=\lim_{n\to\infty}\biggl(n\int_0^{\pi/4}(\tan x)^n\,dx\biggr)$$ I try to this way, $\tan x\ge x$, when $x\in(0,\frac\pi4)$, but this turns out to be $\tan x\ge0$, which is obvious even without ...
5
votes
2answers
105 views

Computing an improper integral with respect to a parameter

I am motivated by this problem.Let us compute an improper integral with respect to a parameter:$$F(x)=\int_{1}^{\infty}\frac{e^{-xy}-1}{y^{3}}dy,\quad x\in[0,\infty).$$ The following is my ...
0
votes
4answers
109 views

$\frac{\pi}2 < \sum_0^\infty \frac{1}{1+n^2} < \frac{3\pi}4 $

Prove that: $\frac{\pi}2 < \sum_0^\infty \frac{1}{1+n^2} < \frac{3\pi}4 $ What I've tried: I solved the improper integral: $\int_0^\infty \frac{1}{1+x^2} = \lim_{b\to \infty} \arctan b ...
0
votes
1answer
34 views

Can the $p$-test be used to determine convergence of integrals of the form $\int^b_a \frac{1}{(c-f(x))^p}dx$?

Can the $p$-test be used to determine convergence of integrals of the form $\int^b_a \frac{1}{(c-f(x))^p}dx$ where $c>f(x)$ for $a\leq x < b$ and $f(b)=c$?
1
vote
2answers
35 views

Get stuck in judging convergence of a improper integral

I get stuck in judging convergence of the following formula:$$\int_0^1{\frac{\sqrt[m]{\ln^2(1-x)}}{\sqrt[n]{x}}}dx$$ $m$ and $n$ are both integers When $x\to0^+$, I can use equivalent infinitesimal ...
0
votes
1answer
25 views

Given the moment generating function of a continuous-type r.v, how to find the p.d.f?

Say for $t<1$: $$M(t) = \frac{1}{(1-t)^2}$$ How to find the p.d.f of the random variable? $$M(t) = E(e^{tx})=\int_{-\infty}^{+\infty}e^{tx}f(x)dx$$ How do we find: $f(x) = xe^{-x}$ on ...
3
votes
2answers
87 views

Can this integral diverge?

Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be a continuous non-increasing bounded function, and suppose $$\limsup_{x\to +\infty}\frac{f(x)}{f(2x)} = +\infty. $$ Can $\int_0^{+\infty} f(x)dx$ diverges? ...
0
votes
2answers
54 views

For what $(a,b) \in R^+$ does $\int^\infty_b (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}})dx$ converge?

For what pairs $(a,b) \in R^+$ does this integral converge? $$ \int\limits^{\infty}_{b} \left (\sqrt{\sqrt{x+a}-\sqrt{x} \vphantom{\sqrt{x}-\sqrt{x-b}}}-\sqrt{\sqrt{x}-\sqrt{x-b}} \right)dx $$
2
votes
1answer
40 views

Fourier transform of $1-\frac{|\tau|}{2T}$

So far I have tried the following: $$\begin{align} \mathscr{F}(f)&=\mathscr{F}\{1-\frac{|\tau|}{2T}\}\\ &=\int_{-\infty}^{+\infty}(1-\frac{|\tau|}{2T})e^{-i\omega\tau}d\tau\\ ...
1
vote
1answer
24 views

Improper integral parametrised in complex variable: when is it holomorphic?

Suppose we are considering the following integral: $$ I(s) = \int_1^\infty t^{s-1}e^{-t\lambda}\;dt $$ where $s \in \mathbb{C}$ and $\lambda > 0$ is a fixed constant. I would like to know when this ...
1
vote
0answers
25 views

Conditions on $f$ to have $ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $ finite?

Suppose that $f$ is a $\mathcal{C}^\infty$ function. $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{f(x)}{(x-y)^2 (y-z)} dz dy dx $$ Which are the conditions on $f$ that makes this integral finite ? ...
2
votes
0answers
63 views

An integral that I cannot simplify.

Good day, esteemed students of mathematics! I have been trying to prove that the convolution of $2q$ Gaussian probability distributions is another $q$ Gaussian probability distribution with the same ...
2
votes
2answers
67 views

Is $\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} dx dy dz$ finite?

My question is in the title : How could I prove that $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} \ \text{d}z \ \text{d}y \ \text{d}x $$ is finite (if it is) ? Thank you by ...
6
votes
2answers
131 views

Integrating $\frac{x^3}{\exp(x)-1}$ from $0$ to $\infty$

While doing Physics and trying to prove the law of Stefan-Boltzmann from Plancks-law one comes to the integral \[ \int_0^\infty \frac{x^3}{\exp(x)-1} \mathrm{d}x=\frac{\pi^4}{15} \] and I would like ...
1
vote
3answers
94 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
-2
votes
0answers
54 views

Integral calculation question [duplicate]

Calculate the following integral: $\int \limits _0 ^\frac \pi 2 \ln (\sin x) \Bbb d x$. We used the substitution $x=2t$ and then used the identity $\sin 2t = 2 \sin t \cos t$ but now we're stuck. ...
1
vote
2answers
29 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
0
votes
3answers
45 views

Integral from minus infinity to plus infinity [duplicate]

I have come across this formula through one of the members of this forum but i dont know how to prove this formula,can someone help me proving this formula. ...
1
vote
0answers
27 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
2
votes
4answers
51 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.