All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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19
votes
3answers
753 views

Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$

Many recent questions have been asked here similar to this integral $$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$ whose "closed form" I cannot seem to figure out. I have ...
19
votes
3answers
422 views

Closed form for integral $\int_{0}^{\pi} \left[1 - r \cos\left(\phi\right)\right]^{-n} \phi \,{\rm d}\phi$

Is there a closed form for $$I_n =\int_{0}^{\pi} \frac{\phi}{(1 - r \cos\phi)^n} \,{\rm d}\phi $$ for $\left\vert\,r\,\right\vert < 1$ real and $n > 0$ integer ? The solution to this integral ...
19
votes
2answers
888 views

Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
19
votes
3answers
757 views

A way of evaluating integrals without doing anything?

The user known as sos440 posted this: $$\begin{align*} \sum_{n=0}^\infty \frac{r^n}{n!} \int_0^\infty x^n e^{-x} \; dx & = \int_{0}^\infty \sum_{n=0}^\infty \frac{(rx)^n}{n!} e^{-x} \; dx = ...
19
votes
1answer
332 views

Closed form for $\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$$
19
votes
1answer
453 views

Weber-type integral

In connection with this answer, I came across the following integral: $$\int_{0}^{\infty} \frac{du}{u} \: \,e^{-t u^2} \frac{J_0(u) Y_0(r u)-J_0(r u) Y_0(u)}{J_0^2(u)+Y_0^2(u)}$$ where $r \gt 1$. I ...
18
votes
3answers
1k views

How do I evaluate this integral $\int_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?

Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi \ln^2 \left({\frac{{1+\sqrt 5 }}{2}}\right).$$ I don't have any idea how to start, ...
18
votes
4answers
384 views

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

I have spent my holiday on Sunday to crack several integral & series problems and I am having trouble to prove the following integral \begin{equation} \int_0^1 ...
18
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
18
votes
4answers
736 views

Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only

I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in ...
18
votes
5answers
1k views

Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am ...
18
votes
4answers
682 views

Prove $\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$

I know that $$\int^\infty_0\frac x{e^x-1}dx=\frac{\pi^2}{6}$$ For substituting $u=2$ into $$\zeta(u)\Gamma(u)=\int^\infty_0\frac{x^{u-1}}{e^x-1}dx$$ However, I suspect that there is an easier proof, ...
18
votes
2answers
2k views

A hard definite integral with trigonometric functions

How could we get a closed form for this one? $$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{x}^{2}}\sqrt{\tan x}\sin \left( 2x \right)\text{d}x}$$
18
votes
6answers
545 views

Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
18
votes
7answers
415 views

Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}$

Hi how can we prove this integral below? $$ I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3} $$ I tried to use $$ I=\int_0^1 \frac{\log^2x}{1-x(1-x)}\mathrm dx $$ and now ...
18
votes
1answer
718 views

Compute $ I_{n}=\int_{-\infty}^\infty \frac{1-\cos x \cos 2x \cdots \cos nx}{x^2}\,dx$

I'm very curious about the ways I may compute the following integral. I'd be very glad to know your approaching ways for this integral: $$ I_{n} \equiv \int_{-\infty}^\infty ...
18
votes
5answers
445 views

Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies ...
18
votes
3answers
551 views

How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$

$$ I:=PV\int_0^\infty \frac{\log\left(\cos^2\left(\alpha x\right)\right)}{\beta^2-x^2} \, \mathrm dx=\alpha \pi,\qquad \alpha>0,\ \beta\in \mathbb{R}.$$ I am trying to solve this integral, I ...
18
votes
4answers
368 views

Evaluating $\int_{0}^{\infty} \frac{x^{3}- \sin^{3}(x)}{x^{5}} \ dx $ using contour integration

EDIT: Instead of expressing the integral as the imaginary part of another integral, I instead expanded $\sin^{3}(x)$ in terms of complex exponentials and I don't run into problems anymore. ...
18
votes
2answers
355 views

A closed form for $\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\ e^{-x\sqrt{2}}}{x}dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the integral $$I=\int_0^\infty\frac{\sin(x)\ ...
18
votes
4answers
761 views

Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
18
votes
4answers
669 views

A “clean” approach to integrals.

Many fields in mathematics start from the "dirty" approach. In calculus we do all sort of $\epsilon$-$\delta$ stuffs, until topology gives an elegant formulation using open sets. A first course in ...
18
votes
4answers
884 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
18
votes
2answers
250 views

A closed form for $\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$

Let $\operatorname{li}x$ denote the logarithmic integral $^{[1]}$$^{[2]}$$^{[3]}$: $$\operatorname{li}x=\int_0^x\frac{dt}{\ln t}$$ and $$I=\int_0^1\frac{\ln(-\ln x)\ ...
18
votes
2answers
1k views

Do Integrals over Fractals Exist?

Given, for example, a line integral like $$ \int_\gamma f \; ds $$ with $f$ not further defined, yet. What happens, if the contour $\gamma$ happens to be a fractal curve? Since all fractal ...
18
votes
2answers
343 views

$\int_0^1f(x)dx=1, \int_0^1xf(x)dx=\frac16$ minimum value of $\int_0^1f^2(x) dx$?

Let $f(x)\geq0$ be a Riemann integrable function, and $$\int_0^1f(x)\,\mathrm dx=1, \int_0^1xf(x)\,\mathrm dx=\frac16.$$ Find the minimum value of $\int_0^1f^2(x)\,\mathrm dx$ Cauchy-Schwarz ...
18
votes
3answers
468 views

Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral \begin{equation} \int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} I have been given two integral questions by my teacher. I cannot answer ...
18
votes
2answers
394 views

Evaluating $\int_0^{\infty} \text{sinc}^m(x) dx$

How do I evaluate $$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$ where $m \in \mathbb{Z}^+$? For $m=1$ and $m=2$, we have the well-known result that this equals $\dfrac{\pi}2$. In ...
18
votes
2answers
342 views

$\int_0^\infty\text{Ci}(x)^3\mathrm dx$

Is there a closed form for this integral: $$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$ where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
18
votes
1answer
693 views

Interesting integral formula

I looked around and found that integrals of the form $$\int_{0}^{\infty} \frac{x^{m-1}}{a+x^n}, a,m,n \in \mathbb{R}, 0<m<n, 0<a$$ seem to occur very often: Just to give a few examples ...
18
votes
3answers
726 views

Why do people interchange between $\int$ and $\sum$ so easily?

One of the things I found curious in many texts is how in certain cases interchange the $\sum$ operator with $\int$. What are the "terms" for such a swap? I understand that integration in the early ...
18
votes
2answers
802 views

Integral $\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$

Please help me to evaluate this integral: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
18
votes
3answers
182 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
18
votes
1answer
342 views

Fourier transform of $\operatorname{erfc}^3\left|x\right|$

(this is a follow-up on my another question) Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^3\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the ...
18
votes
1answer
276 views

Why is there “no analogue of $2i\pi$ in $\mathbf C_p$”?

In his paper Fonctions L p-adiques, Pierre Colmez says: Tate a montré qu'il n'existait pas dans $\mathbf C_p$ d'analogue $p$-adique de $2i \pi$ et donc par conséquent que les périodes $p$-adiques ...
18
votes
2answers
363 views

twist on classic log of sine and cosine integral

I ran across this integral and have not been able to evaluate it. $\displaystyle \int_{0}^{\frac{\pi}{2}}x\ln(\sin(x))\ln(\cos(x))dx=\frac{{\pi}^{2}\ln^{2}(2)}{8}-\frac{{\pi}^{4}}{192}$ I had some ...
18
votes
2answers
764 views

A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

I encountered a strange integral with a strange result. $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$ Believe it or not, the result agrees ...
18
votes
2answers
691 views

A problem for the New Year

What better to start the year than a dazzling integral? $$\int_{0}^{\infty}\left[1+\left(\frac{2013}{x+2013}+\cdots +\frac{2}{x+2}+\frac{1}{x+1}-x\right)^{2014}\,\right]^{-1}\,dx$$ Happy New Year to ...
18
votes
2answers
432 views

Asymptotic analysis of the integral $\int_0^1 \exp\{n (t+\log t) + \sqrt{n} wt\}\,dt$

The integral I'm trying to study is $$ F(n) = \int_0^1 \exp\left\{n(t+\log t)+\sqrt{n}wt\right\}\,dt, \tag{1} $$ where $w$ is a fixed complex number with $\Re(w) < 0$ and $\Im(w) > 0$. As ...
18
votes
1answer
661 views

Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
18
votes
0answers
370 views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method which can find this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
17
votes
4answers
3k views

Why doesn't integrating the area of the square give the volume of the cube?

I had a calculus course this semester in which I was taught that the integration of the area gives the size (volume): $$V = \int\limits_a^b {A(x)dx}$$ But this doesn't seem to work with the square. ...
17
votes
3answers
564 views

Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
17
votes
5answers
1k views

Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...
17
votes
3answers
438 views

Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$

Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that ...
17
votes
3answers
836 views

Evaluate the integral $\int_0^{\infty} \left(\frac{\log x \arctan x}{x}\right)^2 \ dx$

Some rumours point out that the integral you see might be evaluated in a straightforward way. But rumours are sometimes just rumours. Could you confirm/refute it? $$ ...
17
votes
6answers
1k views

Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$

Numerically it seems to be true that $$ \int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}. $$ Any ideas how to prove this?
17
votes
3answers
577 views

Is this definite integral really independent of a parameter? How can it be shown?

I want to find a nice simple expression for the definite integral $$\int_0^\infty \frac{x^2\,dx}{(x^2-a^2)^2 + x^2}$$ Now, I can numerically compute this integral, and it seems to converge to ...
17
votes
2answers
228 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
17
votes
1answer
315 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...