Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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21
votes
6answers
744 views

The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working. $$\begin{align*} u&=f(x)\\ \Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\ \Rightarrow ...
21
votes
2answers
1k views

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

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 ...
21
votes
1answer
335 views

Evaluating :$\int \frac{1}{x^{10} + x}dx$

$$\int \frac{1}{x^{10} + x}dx$$ My solution : $$\begin{align*} \int\frac{1}{x^{10}+x}\,dx&=\int\left(\frac{x^9+1}{x^{10}+x}-\frac{x^9}{x^{10}+x}\right)\,dx\\ ...
21
votes
1answer
394 views

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
21
votes
2answers
521 views

Conjecture: $\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\big(\!\frac{2x}{1+x}\!\big)\,dx\stackrel ?=\frac\pi{5\sqrt2}$

$$\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\left(\frac{2x}{1+x}\right)\,dx\stackrel ?=\frac\pi{5\sqrt2}$$ The integral above comes from the evaluation of the integral ...
21
votes
1answer
359 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
21
votes
3answers
571 views

Evaluating $\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}} \,dx$

How to integrate? $$\int{ \frac{\arctan\sqrt{x^{2}-1}}{\sqrt{x^{2}+x}}}\, dx$$ I have no idea how to do it. Tried to get some information from wiki, but its too hard :|
21
votes
1answer
386 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? $$\int_0^\infty\frac{\sqrt{x+\sqrt{x^2+1}}}{\sqrt{x\phantom{|}}\sqrt{x^2+1}}e^{-x}dx$$
20
votes
10answers
2k views

What is the most efficient method to evaluate this indefinite integral?

$$\int x^5 e^x\,\mathrm{d}x$$ Is there another, more efficient way to solve this integral that is not integration by parts?
20
votes
5answers
353 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
20
votes
4answers
1k 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 ...
20
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: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
20
votes
6answers
607 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 ...
20
votes
5answers
808 views

Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$

I got stuck at the following problem. Let $f\in C([0,\pi/2])$, then compute $$ \lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx $$ Could you suggest a helpful idea?
20
votes
3answers
2k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
20
votes
4answers
476 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 ...
20
votes
3answers
511 views

If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$

If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$ My attempt: Write $x^2+2x\cos{\alpha}+1 = ...
20
votes
2answers
404 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)\ ...
20
votes
3answers
758 views

Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
20
votes
4answers
440 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. ...
20
votes
3answers
280 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 ...
20
votes
2answers
4k 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 ...
20
votes
5answers
615 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 ...
20
votes
3answers
634 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 ...
20
votes
3answers
892 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 ...
20
votes
1answer
405 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
20
votes
2answers
476 views
20
votes
3answers
454 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 ...
20
votes
2answers
1k 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 ...
20
votes
1answer
287 views

A combination integral and series resulting the inverse tangent integral

$\def\Ti{{\rm{Ti}}_2}$I have been able to solve an integral problem, now I tried to use the other method to crack the integral and I have to prove the following expression \begin{equation} ...
20
votes
1answer
819 views

Interesing, hard limit with sum, involving $\pi$

Yesterday I was boring so I decided to derive formula for area of circle with integrals. Very good exercise, I think, because I forgot many, many things about integrals. So I started with: ...
19
votes
7answers
772 views

Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$

Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$ I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated. Thanks.
19
votes
4answers
11k views

Discontinuous derivative.

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is ...
19
votes
3answers
813 views

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

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
19
votes
3answers
584 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 ...
19
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 ...
19
votes
6answers
2k 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?
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
19
votes
8answers
505 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 ...
19
votes
5answers
2k 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 ...
19
votes
5answers
846 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 ...
19
votes
8answers
875 views

Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods

Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$ Thanks.
19
votes
2answers
2k 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 $[\frac{\pi}{2},\pi]$.
19
votes
2answers
464 views

integrate square of $\arctan x$. Tricky

$$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$ I ran across an integral I am having a time solving. The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, ...
19
votes
4answers
589 views

Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$ [duplicate]

I am working on $\displaystyle\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\,dx$ First: I use integrating by part then get $$ x\arctan(\sin x)\Big|_0^\pi-\int_0^\pi \arctan(\sin x)\,dx $$ then I have ...
19
votes
2answers
2k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to ...
19
votes
3answers
350 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 ...
19
votes
4answers
1k 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. ...
19
votes
3answers
449 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
19
votes
4answers
957 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...