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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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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
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
21 votes
3 answers
3k views

Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. [duplicate]

Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$ How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead ...
StationaryTraveller's user avatar
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
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
34 votes
7 answers
17k views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 \left(a+\frac{1}...
Stephen Dedalus's user avatar
29 votes
6 answers
2k views

Real-Analysis Methods to Evaluate $\int_0^\infty \frac{x^a}{1+x^2}\,dx$, $|a|<1$.

In THIS ANSWER, I used straightforward contour integration to evaluate the integral $$\bbox[5px,border:2px solid #C0A000]{\int_0^\infty \frac{x^a}{1+x^2}\,dx=\frac{\pi}{2}\sec\left(\frac{\pi a}{2}\...
Mark Viola's user avatar
  • 181k
39 votes
5 answers
5k views

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

Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
user85798's user avatar
39 votes
8 answers
4k views

Simpler way to compute a definite integral without resorting to partial fractions?

I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$ Is there a simpler way to do this ?
Balaji Rao's user avatar
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
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
45 votes
5 answers
3k 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 \...
zytsang's user avatar
  • 1,533
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
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37 votes
7 answers
8k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...
Jozef's user avatar
  • 7,110
9 votes
7 answers
1k views

Finding $ \int^1_0 \frac{\ln(1+x)}{x}dx$

There is supposed to be a clean solution to the integral below, maybe involving some symmetry $$ \int^1_0 \frac{\ln(1+x)}{x}dx$$ I have tried integration by parts as followed: $\ln(x+1)=u$ ,$\frac{1}{...
bigfocalchord's user avatar
25 votes
3 answers
5k views

Series as an integral (sophomore's dream)

I need help with this exercise. I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$ I think I should use some convergence theorem, but I'm stuck. Thanks a lot!
John Cage's user avatar
  • 253
24 votes
6 answers
6k views

Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$

Suppose $f$ is a continuous, strictly increasing function defined on a closed interval $[a,b]$ such that $f^{-1}$ is the inverse function of $f$. Prove that, $$\int_{a}^bf(x)dx+\int_{f(a)}^{f(b)}f^{-...
Landon Carter's user avatar
21 votes
7 answers
6k views

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

How can I evaluate $$I=\int_{0}^{+\infty}\!e^{-ax^2-\frac b{x^2}}\,dx$$ for $a,b>0$? My methods: Let $a,b > 0$ and let $$I(b)=\int_{0}^{+\infty}e^{-ax^2-\frac b{x^2}}\,dx.$$ Then $$I'(b)=\int_{...
math110's user avatar
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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
7 votes
3 answers
2k views

Integration of a Periodic Function

Problem: $f : \mathbb{R} \to \mathbb{R}$ is a continuous and periodic function with period $T>0.$ Prove that: $$\lim_{n\to +\infty} \int_{a}^{b} f(nx) dx =\frac{b-a}{T} \int_{0}^{T} f(x) ...
giulia85math's user avatar
27 votes
6 answers
4k views

Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $

It's my first post here and I was wondering if someone could help me with evaluating the definite integral $$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$ Thanks in ...
Souvik's user avatar
  • 279
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
50 votes
5 answers
3k views

How to evaluate $\int_0^1\frac{\log^2(1+x)}x\mathrm dx$?

The definite integral $$\int_0^1\frac{\log^2(1+x)}x\mathrm dx=\frac{\zeta(3)}4$$ arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to ...
joriki's user avatar
  • 239k
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
31 votes
8 answers
5k views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan x)\,...
Pranav Arora's user avatar
29 votes
4 answers
9k views

Calculating alternating Euler sums of odd powers

Definition $$\mathbf{H}_{m}^{(n)}(x) = \sum_{k=1}^\infty \frac{H_k^{(n)}}{k^m} x^k\tag{1}$$ We define $$\mathbf{H}_{m}^{(1)}(x) = \mathbf{H}_{m}(x)=\sum_{k=1}^\infty \frac{H_k}{k^m} x^k \tag{2}$$ ...
Zaid Alyafeai's user avatar
19 votes
3 answers
22k views

Is the Riemann integral of a strictly positive function positive?

In the proof here a strictly positive function in $(0,\pi)$ is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a ...
user avatar
16 votes
4 answers
1k views

How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?

Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}\,\mathrm dt=\frac{\pi}{4}.$$ How can I get this formula by hand? Is there any simpler idea than using $u =...
user avatar
42 votes
5 answers
38k views

Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
Pedro's user avatar
  • 6,558
33 votes
4 answers
3k 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. $$\...
Cody's user avatar
  • 1,551
30 votes
10 answers
57k views

Integral of $\frac{1}{(1+x^2)^2}$

I am in the middle of a problem and having trouble integrating the following integral: $$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$ I tried doing partial fractions and got: $$1=A(1+x^2)+B(1+x^2)$$ I ...
user avatar
21 votes
2 answers
17k views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{...
javier's user avatar
  • 261
17 votes
3 answers
7k views

How to solve an definite integral of floor value function?

How do you prove this identity: $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ I'd very much appreciate your help on this one!
Simba's user avatar
  • 445
10 votes
1 answer
452 views

Solving used Real Based Methods: $\int_0^x \frac{t^k}{\left(t^n + a\right)^m}\:dt$

In working on integrals for the past couple of months, I've come across different cases of the following integral: \begin{equation} I\left(x,a,k,n,m\right) = \int_0^x \frac{t^k}{\left(t^n + a\right)^...
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
6 votes
8 answers
6k views

Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$ But I don't know how to explain it. I understand that the derivative of the equation is a ...
Dor's user avatar
  • 1,084
16 votes
2 answers
18k views

Evaluate $\int_0^\pi xf(\sin x)dx$

Let $f(\sin x)$ be a given function of $\sin x$. How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
Steven's user avatar
  • 1,001
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
42 votes
4 answers
2k views

Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?

Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
tomerg's user avatar
  • 1,563
29 votes
6 answers
12k views

An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.

I want to prove: For an integrable function $f(x)$ and periodic with period $T$, for every $a \in \mathbb{R}$, $$\int_{0}^{T}f(x)\;dx=\int_{a}^{a+T}f(x)\;dx.$$ I tried to change the values and ...
Jozef's user avatar
  • 7,110
20 votes
5 answers
2k views

Evaluating $\int_0^1 \frac{\arctan x \log x}{1+x}dx$

In order to compute, in an elementary way, $\displaystyle \int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$ (see Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^...
FDP's user avatar
  • 14k
14 votes
2 answers
2k views

Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $

Prove the following $$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$ where $$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
Zaid Alyafeai's user avatar
21 votes
4 answers
30k views

Why does 'The King Property' of integration work?

Today I learned about something known as the king's property which really helps in solving integrals and I wanted to know why does this property work. I dont know if this terminology is used elsewhere ...
Ashwin Singh's user avatar
11 votes
3 answers
2k views

How to integrate$\int_0^1 \frac{\ln x}{x-1}dx$ without power series expansion

I happen to watch the video here, which gives a solution to the definite integral below using the power series approach. Then answer is $\frac{\pi^2}{6}$, given by: $$\int_0^1 \frac{\ln x}{x-1}dx=\...
student's user avatar
  • 1,830
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
49 votes
9 answers
3k views

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
user153012's user avatar
  • 12.3k
45 votes
10 answers
8k views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable $...
Vladimir Reshetnikov's user avatar
6 votes
2 answers
1k views

Computing the Fourier transform of exponential decay in $\mathbb{R}^2$

I am trying to compute this $\textbf{Fourier transform in}$ $\mathbb{R}^2$ $$ I(\mathbf{k})\equiv\mathcal{F}\big(e^{-a|\mathbf{x}|}\big)(\mathbf{k}) = \int_{\mathbb{R}^2} e^{-a|\mathbf{x}|}e^{i\ \...
ares's user avatar
  • 583
4 votes
5 answers
12k views

How to integrate the product of two or more polynomials raised to some powers, not necessarily integral

This question is inspired by my own answer to a question which I tried to answer and got stuck at one point. The question was: HI DARLING. USE MY ATM CARD, TAKE ANY AMOUNT OUT, GO SHOPPING ...
Anurag Baundwal'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

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