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
255 views

A closed form for $\int_0^\infty e^{-a\,x} \operatorname{erfi}(\sqrt{x})^3\ dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the following parameterized integral $$I(a)=\int_0^\infty ...
19
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 ...
19
votes
2answers
411 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

Does the following series or integral have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx \end{equation} where ...
19
votes
2answers
577 views

Interesting log sine integrals $\int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{7\pi^3}{108}$

Show that $$\begin{aligned} \int_0^{\pi/3} \log^2 \left(2\sin \frac{x}{2} \right)dx &= \frac{7\pi^3}{108} \\ \int_0^{\pi/3}x\log^2 \left(2\sin\frac{x}{2} \right)dx &= ...
19
votes
2answers
396 views

An integral $\int_0^\infty P_s(x-1)\,e^{-x}\,dx$ involving Legendre functions

Let $P_s(x)$ denote the Legendre functions of the $1^{st}$ kind, i.e. the Legendre polynomial generalized to an arbitrary (not necessarily integer) order $s$. It can be expressed using the ...
19
votes
1answer
292 views

A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

I am stuck with this integral: $$\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx,$$ where $J_0$ is the Bessel function of the first kind. Is it possible to express this ...
19
votes
3answers
772 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
2answers
901 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
760 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
340 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? $$\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$$
18
votes
2answers
1k views

Is indefinite integration non-linear?

Let us consider this small problem: $$ \int0\;dx = 0\cdot\int1\;dx = 0\cdot(x+c) = 0 \tag1 $$ $$ \frac{dc}{dx} = 0 \qquad\iff\qquad \int 0\;dx = c, \qquad\forall c\in\mathbb{R} \tag2 $$ These are two ...
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
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
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
693 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
555 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
426 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
728 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
486 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
557 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
2answers
363 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
387 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
4answers
859 views

Interesting integral related to the Omega Constant/Lambert W Function

I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am sorry. ...
18
votes
4answers
672 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
2answers
3k views

Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider ...
18
votes
4answers
894 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
258 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
5answers
571 views

Evaluating $\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx$

Evaluate the following limit: $$\lim \limits_{n\to \infty}\,\,\, n\!\! \int\limits_{0}^{\pi/2}\!\! \left(1-\sqrt [n]{\sin x} \right)\,\mathrm dx $$ I have done the problem . My method: First I ...
18
votes
2answers
349 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
512 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
399 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
344 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
720 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
734 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
810 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
188 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
422 views

Integral $\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}$

Please help me prove the following identity: $$\int_0^\infty \frac{\sin x}{\cosh ax+\cos x}\frac{x}{x^2-\pi^2}dx=\tan^{-1}\left(\frac{1}{a}\right)-\frac{1}{a}\quad a>0$$ This integral is from the ...
18
votes
1answer
346 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
2answers
969 views

Integral $\int_1^\infty\dfrac{dx}{1+2^x+3^x}$

Can the integral $$\int_1^\infty\dfrac{dx}{1+2^x+3^x}$$ be given in closed form? This question arises naturally when I considered doing integrals. What makes an integral hard? Well, the integrand, of ...
18
votes
1answer
284 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
378 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
785 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
718 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
1answer
335 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
18
votes
2answers
438 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
663 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
2answers
444 views

Ramanujan style nested differential Equation

So I was exploring some math the other day... and I came across the following neat identity: Given $y$ is a function of $x$ ($y(x)$) and $$ y = 1 + \frac{\mathrm{d}}{\mathrm{d}x} \left(1 + ...
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
7answers
955 views

A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$?

I am currently trying to show that $\int_{-\infty}^\infty \cos(x^2) \, \mathrm dx = \sqrt{\frac{\pi}{2}}$ and the last integral I have to evaluate is $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...