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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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
785 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
3answers
699 views

Prove $\int_0^\infty \frac{\ln(\tan^2 (ax))}{1+x^2}dx = \pi\ln(\tanh(a))$

How would I prove $$\displaystyle\int_0^\infty \frac{\ln(\tan^2 (ax))}{1+x^2}dx = \pi\ln(\tanh(a))?$$
17
votes
7answers
376 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 ...
17
votes
3answers
571 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
3answers
514 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 ...
17
votes
2answers
1k views

Calculating $\int_{\pi/2}^{\pi}\frac{x\sin{x}}{5-4\cos{x}}\,\mathrm dx$

Calculate the following integral:$$\int_{\pi/2}^{\pi}\frac{x\sin{x}}{5-4\cos{x}}\,\mathrm dx$$ I can calculate the integral on $[0,\pi]$,but I want to know how to do it on $[\frac{\pi}{2},\pi]$.
17
votes
2answers
238 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)\ ...
17
votes
4answers
812 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 ...
17
votes
4answers
626 views

Is there a way to prove $\int {x^n e^x dx} = e^x \sum_{k = 0}^n {( - 1)^k \frac{{n!}}{{(n-k)!}}x^{n-k} } + C$ combinatorially?

In How to integrate $\int x^n e^x dx$?, it is shown that $$\int {x^n e^x dx} = e^x \sum_{k = 0}^n ( - 1)^k \frac{n!}{(n-k)!}x^{n-k} + C.$$ Since $\frac{n!}{(n-k)!}$ is $P(n,k)$, the number of ...
17
votes
3answers
312 views

Closed form for n-th anti-derivative of $\log x$

Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
17
votes
2answers
327 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?
17
votes
2answers
369 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 ...
17
votes
4answers
652 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
17
votes
1answer
383 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 ...
17
votes
2answers
282 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove\begin{equation} \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln(\cos x))^2 \right) \, dx=\pi\ln\ln2 \end{equation} I don't know how to answer it. When I asked this integral to my brother, ...
17
votes
2answers
720 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 ...
17
votes
1answer
317 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) ...
17
votes
1answer
439 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
16
votes
6answers
985 views

Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$?

Which is the easiest way to evaluate $\int \limits_{0}^{\pi/2} (\sqrt{\tan x} +\sqrt{\cot x})$? I have reduced this problem to $$ 2\int_0^{\pi/2} \sqrt{\tan x} \ dx$$ but now, evaluating this ...
16
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 ...
16
votes
3answers
520 views

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

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
16
votes
6answers
560 views

Show that $\int_{0}^{\infty }\frac {\ln x}{x^4+1}\ dx =-\frac{\pi^2 \sqrt{2}}{16}$

I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that $$ ...
16
votes
4answers
468 views

How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
16
votes
5answers
758 views

Evaluation of the integral $\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$

How can I evaluate the integral $$\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$$ I tried manipulating the known integral $$\int_0^1 \frac{\ln(1 - x)}{x}dx = -\frac{\pi^2}{6}$$ but couldn't do anything with ...
16
votes
2answers
427 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
16
votes
3answers
254 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...
16
votes
4answers
723 views

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

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
16
votes
4answers
774 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. ...
16
votes
2answers
334 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)\ ...
16
votes
1answer
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 ...
16
votes
5answers
536 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 ...
16
votes
3answers
500 views

Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$

I wonder what kind of functions satisfy $$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$ I suppose all functions must be continuous.
16
votes
1answer
228 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
16
votes
1answer
298 views

Prove the following integral inequality

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
16
votes
2answers
565 views

Showing that $ \int_{0}^{\pi/2}\frac{1}{\sqrt{\sin{x}}}\;{dx}=\int_{0}^{\pi/2}\frac{2}{\sqrt{2-\sin^2{x}}}\;{dx}?$

How can we show that $ \displaystyle \int_{0}^{\pi/2}\frac{1}{\sqrt{\sin{x}}}\;{dx}=\int_{0}^{\pi/2}\frac{2}{\sqrt{2-\sin^2{x}}}\;{dx}? $ It feels like it should be simple, but I've tried many things ...
16
votes
2answers
866 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 ...
16
votes
5answers
362 views

Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$

Question: Find the integral $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ my try: since $$I=2\int_{0}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ then I can't. I ...
16
votes
3answers
474 views

A generalized (MacLaurin's) average for functions

The average value of a function $y=f(x)$, on an interval $[a,b]$, is ${1\over {b-a}}\int_a^b f(t)dt$. This of course relates to the arithmetic average. It is easy to see that a corresponding formula ...
16
votes
1answer
284 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? $$\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$$
16
votes
1answer
398 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 ...
16
votes
1answer
399 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 + ...
15
votes
8answers
1k views

$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral

Compute the following integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$ Any hint, suggestion is welcome. Thanks.
15
votes
6answers
2k views

Please show $\int_0^\infty x^{2n} e^{-x^2}\mathrm dx=\frac{(2n)!}{2^{2n}n!}\frac{\sqrt{\pi}}{2}$ without gamma function?

Prove: $$\int_0^\infty x^{2n} e^{-x^2}\mathrm dx=\frac{(2n)!}{2^{2n}n!}\frac{\sqrt{\pi}}{2}$$ Thanks!
15
votes
7answers
905 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 $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...
15
votes
4answers
1k views

$\int^{1}_{0} f^{-1} = 1 - \int^1_0 f$

One more from hard to believe facts, which I'm curious why are true. Let $f : [0,1] \rightarrow [0,1] $ is a continuous, monotonically increasing and surjective function Then $$\int^{1}_{0} f^{-1} ...
15
votes
3answers
636 views

Showing that $ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dt=\ln2 $

I would like to show that $$ \int_{0}^{1} \frac{x-1}{\ln(x)} \mathrm dt=\ln2 $$ What annoys me is that $ x-1 $ is the numerator so the geometric power series is useless. Any idea?
15
votes
8answers
3k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
15
votes
6answers
829 views

Compute $\lim\limits_{n\to\infty} \int_{0}^{2\pi} \cos x \cos 2x\cdots \cos nx \space{dx}$

Compute the following limit: $$\lim_{n \to \infty}\int_{0}^{2\pi}\cos\left(x\right)\cos\left(2x\right)\ldots \cos\left(nx\right)\,{\rm d}x$$ Today I was working on a W. L. Putnam competition's ...
15
votes
3answers
468 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} f\left( x - ...