Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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58
votes
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2k 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 ...
43
votes
1answer
2k views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method to evaluate this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
35
votes
1answer
516 views

Prove that $\frac{1}{\phi}<\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx< \frac{24+\sqrt{2}}{41} $

I'm sure that's a coincidence, but the Laplace transform of $1/\Gamma(x)$ at $s=1$ turns out to be pretty close to the inverse of the Golden ratio: $$F(1)=\int_0^\infty \frac{e^{-x}}{\Gamma(x)} dx=0....
25
votes
0answers
829 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{\...
19
votes
0answers
897 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}\,\mathrm{d}x.$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
18
votes
0answers
324 views

$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{x_1^{505}+\cdots +x_n^{505}}{x_1^{2020}+\cdots +x_n^{2020}}dx_1\cdots dx_n$

Evaluate this multiple integral inside a limit: $$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{ \sum _{k=1}^{n}x_k^{505}}{\sum_{k=1}^{n}x_k^{2020}} \mathrm d x_1\cdots \...
16
votes
0answers
515 views

How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$

How to prove$$\int_0^1x\ln^2(1+x)\ln\left(\frac{x^2}{1+x}\right)\frac{dx}{1+x^2}=-\frac{7}{32}\cdot\zeta{(3)}\ln2+\frac{3\pi^2}{128}\cdot\ln^22-\frac{1}{64}\cdot\ln^42-\frac{13\pi^4}{46080}$$ The ...
15
votes
0answers
286 views

What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?

As in this post, define the ff: $$K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$ $$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$ $$K_4(k)={\tfrac{\...
14
votes
0answers
283 views

On the relationship between $\Re\operatorname{Li}_n(1+i)$ and $\operatorname{Li}_n(1/2)$ when $n\ge5$

Motivation $\newcommand{Li}{\operatorname{Li}}$ It is already known that: $$\Re\Li_2(1+i)=\frac{\pi^2}{16}$$ $$\Re\Li_3(1+i)=\frac{\pi^2\ln2}{32}+\frac{35}{64}\zeta(3)$$ And by this question, ...
14
votes
0answers
975 views

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\...
13
votes
0answers
342 views

Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
13
votes
1answer
748 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}...
12
votes
0answers
652 views

Omega Constant Integral

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to ...
12
votes
1answer
228 views

Volume of the intersection of two simplexes

Let $S_n$ be the interior of the unitary $n$-simplex, i.e $ S_n =\{{\bf x} \in \mathbb{R}^n \mid x_i\ge0 \wedge \sum_{i=1}^n x_i\le1\}$ Let $T_n({\bf y})$ be the reversed simplex with origin at ${\...
11
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0answers
395 views

Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$

In This Answer, I wrote "It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$." For completeness, I've included the "...
11
votes
0answers
254 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
10
votes
0answers
424 views

Proof of a non-trivial non-linear relation for $\int_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx$?

Mordell type integrals are integrals of the following form $$ \phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1} $$ G.N. Watson in his paper ...
10
votes
1answer
286 views

$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\frac{m_1 z}{a} + \frac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz.$

Question: How to find the closed-form solution for the given integral? $$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\dfrac{m_1 z}{a} + \dfrac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz,$$ where $k, a, b,...
10
votes
0answers
165 views

Integral of $\int_0^{\infty} \ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm 1}dx$

so I have this integral to try and evaluate $$(*)=\int_0^{\infty} ln\left|\frac{x+A}{x+B}\right|\frac{x}{e^{C x}\pm1}dx$$ So far, I have managed to evaluate a very similar integral $$\int_0^{\infty}...
10
votes
0answers
635 views

Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
10
votes
1answer
168 views

Double integral - transformation

I'm trying to calculate $$\iint_{\Omega } e^{(x+y^2)^{3/2}} \,\mathrm{d}A,$$ where $$\Omega =\{x,y>0 : x+y\leq 2\}. $$ Not sure where to go with it. I need to find a transformation and then ...
10
votes
1answer
371 views

An extremely mysterious integral: $\int_0^1 \frac{k \tan^{-1}(t)}{k^2 + t^2}\mathrm d t$

$$f(n) = \int_0^1 \frac{n \tan^{-1}(t)}{n^2 + t^2}\mathrm d t \tag{n > 2}$$ Introduction: This is one of the most beautiful and mysterious integrals I've every encountered. It's very simple, but ...
10
votes
0answers
1k views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
10
votes
0answers
1k views

Difficult integral for a marginal distribution

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral: $$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} \frac{\sqrt{\frac{2}{\...
9
votes
0answers
402 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in his ...
9
votes
0answers
473 views

Mixed Bessel Function integral $\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\mathrm{d}z$

A tricky integral I have been working on, and probably doesn't have a solution in terms of known functions, is: $$\int_{0}^{\infty} e^{- \lambda \left(\sqrt{(z+a)^2+b^2}+\sqrt{(z+c)^2+d^2}~\right)}\...
8
votes
0answers
375 views

Help with an unusual integral

Let $n$ be a positive integer, and $s\in \mathbb{C}\;,\Re(s)>0$. I want to compute the integral : $$\int_{0}^{\infty}\sin\left(2\pi ne^{x}\right)\left[\frac{s}{e^{sx}-1}-\frac{1}{x}\right]dx$$ I ...
8
votes
0answers
139 views

Average value of $\ln(1+e^x)$ when $x$ is normally distributed

Does the following integral admit a closed form answer: $$\int_{-\infty}^\infty\mathrm d x \exp\left(-\frac{(x-\mu)^2}{2\nu}\right) \ln(1+e^x)$$ where $\nu>0$ and $\mu$ are finite real parameters....
8
votes
0answers
110 views

A special integral related to Beta function

Consider the integral $$ I(a,b) = \int_0^\infty \big(z^a-(z-1)_+^a\big)z^{-b}dz $$ for some $a\in \mathbb R$, $b\in (a,a+1)$ (here $z_+ = \max(z,0)$). When $a>0$, we can write $$ I(a,b) = a\int_0^\...
8
votes
0answers
211 views

Integrals involving powers and beta (or hypergeometric) function

I have the three following integrals, very similar the one to the others, $$I_1^{(p)}(N)\equiv\frac{1}{2^{N+p}}\int_0^1(1+t)^{N-1}(1-t)^pB\left(\frac{1}{t+1};N+p+1,N\right)\text{d}t$$ $$I_2^{(p)}(...
8
votes
0answers
277 views

Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$

This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while. ...
8
votes
0answers
132 views

Equivalence class of definite integrals

Let's assume we have a smooth function $f(x):[a,b]\to \mathbb{R}$ so that the integral $$\int_a^b f(x) dx$$ is finite. By performing various changes of variables, we can derive a large (infinite?) ...
8
votes
0answers
1k views

Exact values of error function

The error function is defined as $$\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} \, dt.$$ We know that the Gaussian integral is $$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}.$$ ...
8
votes
1answer
195 views

How to integrate $\frac{x^{2}\log {\sin x}}{1+x^{6}}$

I recently stumbled upon a question $$\int_0^{\infty}\frac{x^{m-1}\log^{a}x}{1+x^n}dx$$ I was able to evaluate it,but I am curious if there exists a closed form for, $$\int_0^{\pi/2}\frac{x^{2}\log{\...
7
votes
0answers
103 views

Integrating $x^{x^x}$

Although one cannot find an elementary antiderivative of $f(x)=x^x$, we can still give a series representation for $\int_0^1 x^x dx$, namely: $$I_1=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^n}=0.78343\...
7
votes
0answers
62 views

Closed form of $\int_0^1\frac{W_0(-t/e)}{W_{-1}(-t/e)} \,dt$

$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Ei{\operatorname{Ei}}$ Is there a known closed form for ...
7
votes
0answers
60 views

Closed form for the integral $\int_{-\pi/2}^{\pi/2} W(\sec(\varphi)) d\varphi$?

I'm interested in the integral $$I=\int_{-\pi/2}^{\pi/2} W(\sec(\varphi)) d\varphi$$where $W$ is the Lambert W function, defined such that $W(x)e^{W(x)}=x$. The numerical value for $I$, as yielded by ...
7
votes
0answers
173 views

The sine cardinal function and $F_1 = F_2 = F_3 = F_4 = F_5 = F_6 = 0$

Define the function, $$F_n=\frac12-\int_0^\infty \frac{\sin^n x}{x^n}\,dx+\sum_{x=1}^\infty \frac{\sin^n x}{x^n}\tag1$$ where $\rm{sinc}^n(x)=\frac{\sin^n x}{x^n}$ is the sine cardinal function. We ...
7
votes
1answer
491 views

Infinite summation formula for modified Bessel functions of first kind

I was trying to find a closed form for the integral $$4\int_0^{\pi/2} t \, I_0(2\kappa\cos{t}) dt \; ,$$ where $$I_{\alpha}(z) := i^{-\alpha}J_{\alpha}(iz) = \sum_{m=0}^{\infty}\frac{\left(\frac{z}{...
7
votes
1answer
209 views

Is there a closed form for the integral $\int_0^\infty \frac{e^{-x^2} I_0 \left(\beta x \right) d x}{\sqrt{ \alpha^2+x^2}}$

I encountered this integral in my work, and it would be really convenient if it had a closed form in terms of any known special functions (which Mathematica could handle): $$J(\alpha,\beta)=\int_0^\...
7
votes
0answers
167 views

Methods to solve $\int_{0}^{\infty} \frac{\cos\left(kx^n\right)}{x^n + a}\:dx$

Spurred on by this question, I decided to investigate for different functions on the numerator. Here, I went from $\exp(..)$ to $\sin(..) / \cos(..)$. I initially thought I could modify the result ...
7
votes
0answers
244 views

How prove this integral $\int_0^1\frac{(\arctan{x})^2\ln({x+1/x+2})}{(1+x)^2}dx$

$$I=\int_0^1\frac{(\arctan{x})^2\ln({x+1/x+2})}{(1+x)^2}dx=-\dfrac{\pi^3}{96}+\dfrac{5\pi}{16}\ln^22-\dfrac{\pi}{4}G-G+\dfrac{\pi}{2}\ln2+\dfrac{7}{16}\zeta(3)$$ Where G is the Catalan's Constant. ...
7
votes
0answers
342 views

Is there a closed form for $\,_4 F_3(1,1,1,3; 3/2,5/2,5/2;1)$?

A semi-algebraic generalization of the Steiner surface has appeared, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...
7
votes
1answer
116 views

three integrals sum to a $_{3}F_{2}$ value

Let $K(x)$ and $E(x)$ denote complete elliptic integrals of the first and second kind. Let $$A=\frac{1024}{9\pi^{3}} \int\limits_{0}^{\infty} \,\frac{t\left( 8t^{4}+8t^{2}-1\right) E\left( i\,t\...
7
votes
0answers
236 views

definite integral of elliptic integral of first kind

The signal-to-noise ratio of a Hall-effect magnetic sensor is proportional to $$ H(f,p)=\frac{I_1 (f,p)}{\sqrt{KK'(\frac{1-f}{1+f})} \sqrt{KK'(\frac{1-p}{1+p})}} $$ with $KK'(x)=K(x)K'(x)$ and $K'(x)=...
7
votes
1answer
428 views

Old & cool integral $\int_0^{\pi} \sin^{b-1}(x) \sin(a x) \ dx=\frac{\pi \sin(a \pi/2)}{2^{b-1}b B\left(\frac{b+a+1}{2},\frac{b-a+1}{2}\right)}$

Here is an integral that appears in the table of integrals by Gradshtein and Ryzhik, it was also studied by Ramanujan (not sure his original solution was found - it seems it doesn't appear in any of ...
7
votes
0answers
189 views

Calculation of $\int_{0}^1 \frac{\sin(\ln^4(1-x))}{x}~dx$

$$I=\int_0^1 \frac{\sin(\ln^4 (1-x))}{x}dx$$ What is the closed-form evaluation of this integral? I honestly do not have a single clue how to solve this. (There is no application, but it is out of ...
7
votes
0answers
118 views

How to solve this definite Integral containing $E_{1}${.}!

The integral is: $$\int_{N}^{\infty}\frac{E_{1}(cz+d)}{az+b}e^{-pz}dz$$ where, $E_{1}${.} is the exponential integral, and $$a>0,\ b>0,\ c>0,\ d>0,\ p>0,\ N>0.$$ This is similar ...
7
votes
0answers
209 views

The minimum of $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx$ is attained for $k=n$

I have the following conjecture: ``For each given $n\in\mathbb{N},\ n\ge 2$ the minimum of the sequence of integrals $I_{n,k}=\int_0^{2\pi}\sqrt{3+2\cos(nx)+2\cos(kx)+2\cos(nx+kx)}dx,\ k=1,2,\dots,n$ ...
7
votes
0answers
391 views

Can we interchange the Integral and Summation when a limit is $\infty$?

I was trying to Evaluate the Integral: $$\Large{I=\int_1^{\infty} \frac{\ln x}{x^2+1} dx}$$ $$\color{#66f}{{\frac{1}{x^2+1} = \frac{1}{x^2\left(1+\frac{1}{x^2}\right)}=\frac{1}{x^2}\cdot \frac{1}{1+...

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