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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25
votes
3answers
610 views

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
1
vote
2answers
38 views

Convergence of $\int_2^{\infty}f(x)\,dx$ with a given condition

Let , $f$ be continuous function on $[2,\infty)$ and $\displaystyle\lim_{x\to \infty}x(\log x)^pf(x)=A$ , where $A$ is a non-zero finite number.. Then $\displaystyle\int_2^{\infty}f(x)\,dx$ is (A) ...
4
votes
1answer
167 views

Calculate the Gauss integral without squaring it first

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 $2-$dimensional integral in the plane and integrate it in ...
10
votes
8answers
324 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
6
votes
1answer
204 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 ...
1
vote
4answers
119 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 ...
4
votes
5answers
73 views

dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral $ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $ by integration by parts I get; $ \displaystyle ...
1
vote
1answer
27 views

Identical Summation and Integration of specific functions

A strange coincidence which I discovered recently, is that $$\int_{-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}dx = ...
14
votes
3answers
475 views

Under what conditions is integrating over a series expansion valid for an improper integral?

On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...
2
votes
0answers
35 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...
-1
votes
3answers
60 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
37 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 ...
8
votes
3answers
208 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
9
votes
3answers
222 views

Compute $\int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}\,dx$

Compute the definite integral $$ \int_0^\infty\frac{x^a-x^b}{(1+x^a)(1+x^b)}\,dx $$ where $a,b\in\mathbb{R}$. My Attempt: Let $x=\frac{1}{t}$ so that $dx=-\frac{1}{t^2}\,dt$. ...
7
votes
3answers
436 views

Integral $\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx$

How can I show that $$\int_{-\infty}^{\infty}\frac{e^{r \arctan(ax)}+e^{-r \arctan(ax)}}{1+x^2}\cos \left( \frac{r}{2}\log(1+a^2x^2)\right)dx = 2\pi \cos \left( r\log(1+a)\right)$$ where $a \in ...
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
-1
votes
1answer
115 views
0
votes
0answers
15 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
12 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 ...
127
votes
2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$

Evaluate the following integral $$ \tag1\int_{0}^{\pi/2}\frac1{(1+x^2)(1+\tan x)}\,dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,dx = ...
2
votes
3answers
258 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
191 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 ...
1
vote
1answer
49 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} \ln(2+x)dx$$ Don't know where to start..
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 ...
26
votes
4answers
526 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 ...
35
votes
4answers
1k views

What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?

/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Worlfram alpha gives an indefinite form in terms of the logarithmic integral ...
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
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.
6
votes
4answers
148 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 ...
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 ...
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, ...
6
votes
0answers
148 views

How to compute or simplify this integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
0
votes
1answer
63 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 ...
27
votes
4answers
967 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} ...
0
votes
0answers
35 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 ...
3
votes
2answers
109 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}\, ...
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 ...
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 ...
4
votes
4answers
86 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 ...
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 ...
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 ...
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 ...
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 ...
78
votes
16answers
11k views
12
votes
5answers
374 views

Show $\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$ for $t \gt 0$

The problem is to show $$\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$$ for $t \gt 0$. I'm pretty stuck. I thought about integration by parts and couldn't get anywhere with the integrand in ...
2
votes
3answers
65 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
4answers
66 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) + ...