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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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593 votes
14 answers
380k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
Laila Podlesny's user avatar
313 votes
21 answers
58k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
user3002473's user avatar
  • 8,974
278 votes
5 answers
25k views

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

Evaluate the following integral $$ \tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\Bbb dx $$ My Attempt: Letting $x=\frac{\pi}{2}-x$ and using the property that $$ \int_{0}^{a}f(x)\,\Bbb dx =...
juantheron's user avatar
  • 53.5k
274 votes
32 answers
134k views

Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$ Well, can ...
user avatar
254 votes
11 answers
22k views

"Integral milking": working backward to construct nontrivial integrals

I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others ...
Franklin Pezzuti Dyer's user avatar
201 votes
4 answers
29k views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
JessicaK's user avatar
  • 7,685
197 votes
2 answers
9k views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
Sangchul Lee's user avatar
181 votes
6 answers
13k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: $$\int_0^1\ln\left(1+\frac{\ln^2x}{...
Vladimir Reshetnikov's user avatar
179 votes
6 answers
9k views

Symmetry of function defined by integral

Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for ...
Ron Gordon's user avatar
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176 votes
4 answers
41k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\...
Shobhit Bhatnagar's user avatar
135 votes
11 answers
41k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: $$\begin{...
Oksana Gimmel's user avatar
135 votes
2 answers
6k views

How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$

How can one prove that $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
larry's user avatar
  • 1,499
132 votes
5 answers
11k views

Help find hard integrals that evaluate to $59$?

My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$. I want to try and make a definite integral that equals $59$. So ...
121 votes
5 answers
41k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
Cleo's user avatar
  • 21.6k
118 votes
13 answers
17k views

Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
Martin Gales's user avatar
  • 6,898
118 votes
4 answers
12k views

Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$

Compute the following integral \begin{equation} \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]\, ...
Anastasiya-Romanova 秀's user avatar
116 votes
11 answers
10k views

Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: $$\...
Pedro's user avatar
  • 123k
106 votes
8 answers
48k views

Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$

Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
user 1591719's user avatar
  • 44.3k
96 votes
2 answers
7k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. Can ...
Vladimir Reshetnikov's user avatar
96 votes
4 answers
11k views

Do numbers get worse than transcendental?

Mathematicians have come up with many ways of classifying how "exotic" some numbers are. For example, the most ordinary numbers are the natural "counting" number, and the next most exotic numbers are ...
Franklin Pezzuti Dyer's user avatar
88 votes
1 answer
4k views

Can someone explain this integration trick for log-sine integrals?

I was working on this rather challenging log-sine integral: $$ \int_{0}^{2\pi}x^{2}\ln^{2}\left(2\sin\left(x \over 2\right)\right)\,{\rm d}x = {13\pi^{5} \over 45} $$ The upper limit is a waiver from ...
Cody's user avatar
  • 14.1k
87 votes
4 answers
6k views

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

While browsing on Integral and Series, I found a strange integral posted by @Sangchul Lee. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind ...
Venus's user avatar
  • 11k
83 votes
2 answers
4k views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
Olivier Oloa's user avatar
81 votes
1 answer
4k views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools or ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$ EDIT: It took a while since I made this post. I'll ...
user 1591719's user avatar
  • 44.3k
70 votes
5 answers
9k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I ...
Frida Mauer's user avatar
  • 1,249
69 votes
4 answers
22k views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for }a>0,\,b>0}\...
Vladimir Reshetnikov's user avatar
69 votes
11 answers
4k views

Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?

I know the following is not right, but what is the problem. So we want to calculate $$ \int_0^{\pi} \sin(x) \; dx $$ If one does a substitution $u = \sin(x)$, then one gets $$ \int_{\sin(0) = 0}^{\...
John Doe's user avatar
  • 3,283
69 votes
5 answers
4k views

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
Laila Podlesny's user avatar
69 votes
4 answers
4k views

A nasty integral of a rational function

I'm having a hard time proving the following $$\int_0^{\infty} \frac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \frac{\pi}{2}.$$ Mathematica has no ...
user avatar
67 votes
2 answers
3k views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}...
Olivier Oloa's user avatar
66 votes
7 answers
6k views

$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$ on 2015 MIT Integration Bee

So one of the question on the MIT Integration Bee has baffled me all day today $$\int_{0}^{\frac{\pi}{4}}\frac{\tan^2 x}{1+x^2}\text{d}x$$ I have tried a variety of things to do this, starting with ...
Teh Rod's user avatar
  • 3,108
66 votes
2 answers
4k views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also didn'...
user153012's user avatar
  • 12.3k
66 votes
1 answer
3k views

Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$ Or equivalently: $$\boxed{\...
Zacky's user avatar
  • 28.1k
63 votes
19 answers
21k views

How do I convince my students that the choice of variable of integration is irrelevant?

I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ...
62 votes
3 answers
4k views

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

Does $$~\displaystyle{\int}_0^1\frac{\text{arctanh }x}{\tan\left(\dfrac\pi2~x\right)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$$ possess a closed form expression ? This recent post, in ...
Lucian's user avatar
  • 48.5k
62 votes
4 answers
5k views

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

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
Shobhit's user avatar
  • 1,080
60 votes
2 answers
7k views

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

Regarding this problem, I conjectured that $$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi \...
Sangchul Lee's user avatar
59 votes
9 answers
7k views

How to evaluate $\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$

Find the value of $$I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$$ We have the information that $$J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)\ \mathrm dx=\dfrac{\pi^2}{8}...
math110's user avatar
  • 93.6k
57 votes
7 answers
10k 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 $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral ...
YoniY's user avatar
  • 581
57 votes
8 answers
3k views

Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$ [closed]

Evaluate $$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$$
user 1591719's user avatar
  • 44.3k
57 votes
1 answer
4k views

Evaluating $\int_0^\infty \frac{dx}{\sqrt{x}[x^2+(1+2\sqrt{2})x+1][1-x+x^2-x^3+...+x^{50}]}$

My brother's friend gave me the following wicked integral with a beautiful result \begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \bigg[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] \bigg[1-x+x^2-x^...
Anastasiya-Romanova 秀's user avatar
56 votes
7 answers
5k 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.
lar49's user avatar
  • 781
55 votes
5 answers
5k views

A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Consider the following integral: $$\mathcal{I}=\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using ...
Vladimir Reshetnikov's user avatar
55 votes
1 answer
2k views

To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

$$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+a^2}\sqrt{x^2+b^2}}$$ To use Landen's transformation $$f(a,b)=\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt{x^2+(\frac{a+b}{2})^2}\sqrt{x^2+ab}}$$ $...
Mathlover's user avatar
  • 10.1k
55 votes
4 answers
2k views

Prove/disprove $(\int_0^{2 \pi} \!\!\cos f(x) \, d x)^2+(\int_0^{2 \pi}\!\!\! \sqrt{(f'(x))^2+\sin ^2 f(x)} \, dx)^2\ge 4\pi^2$

Let $f(x)$ be a differentiable function on $[0,2\pi]$ s.t. $0\leq f(x)\leq 2\pi$ and $f(0)=f(2\pi)$. Prove or disprove that $$ \left(\int_0^{2 \pi} \cos f(x) \,d x\right)^2+\left(\int_0^{2 \pi} \sqrt{(...
FFjet's user avatar
  • 5,054
54 votes
1 answer
2k views

A curious equality of integrals involving the prime counting function?

This post discusses the integral, $$I(k)=\int_0^k\pi(x)\pi(k-x)dx$$ where $\pi(x)$ is the prime-counting function. For example, $$I(13)=\int_0^{13}\pi(x)\pi(13-x)dx = 73$$ Using WolframAlpha, the ...
Tito Piezas III's user avatar
54 votes
2 answers
4k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos 2\...
Zakharia Stanley's user avatar
53 votes
5 answers
2k views

How to evaluate $\int_0^\infty\operatorname{erfc}^n x\ \mathrm dx$?

Let $\operatorname{erfc}x$ be the complementary error function. I successfully evaluated these integrals: $$\int_0^\infty\operatorname{erfc}x\ \mathrm dx=\frac1{\sqrt\pi}\tag1$$ $$\int_0^\infty\...
Hanna K.'s user avatar
  • 1,376
53 votes
3 answers
4k views

Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$

Prove the following integral $$I=\int\limits_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\dfrac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$$ This integral result ...
math110's user avatar
  • 93.6k
52 votes
3 answers
3k views

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

I encountered this integral in my calculations: $$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx=2\int_0^\infty\frac{x\log\left(1+\frac{\pi^2}{4\,x^2}\right)}{e^x-...
Oksana Gimmel's user avatar

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