5
votes
1answer
102 views

Is there a theory of integration in elementary terms for definite integrals?

Let's call a real number explicit if it can be expressed starting from integers by using arithmetic operations, radicals, exponents, logarithms, trigonometric and inverse trigonometric functions. For ...
12
votes
2answers
358 views
+100

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
6
votes
1answer
71 views

Closed form for integral of integer powers of Sinc function

(Edit: Thank you Vladimir for providing the references for the closed form value of the integrals. My revised question is then to how to derive this closed form.) For all $n\in\mathbb{N}^+$, ...
2
votes
1answer
62 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $$I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta,$$ where $a$ and $b$ are real numbers. If we ...
2
votes
1answer
96 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
10
votes
0answers
163 views
6
votes
0answers
118 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
16
votes
2answers
328 views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
11
votes
2answers
353 views

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

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 ...
5
votes
0answers
124 views

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx$

Please help me to evaluate this integral: $$I={\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ dx.\tag1$$ Mathematica could not evaluate it in a closed form. A numerical ...
4
votes
2answers
155 views

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
5
votes
2answers
203 views

A closed form of $\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left(\frac{1}{x}\right)}{x^2-x+1}dx$$ I've tried substitution $y=\frac{1}{x}$ and $e^y=\frac{1}{x}$, but those didn't ...
4
votes
1answer
109 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
1
vote
2answers
47 views

closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$

$$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$ my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function so could some one help ?
2
votes
0answers
60 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
2
votes
2answers
111 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
7
votes
3answers
204 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
7
votes
1answer
144 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
20
votes
4answers
584 views

Evaluate $\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx$

Let $n\ge2$ be an integer , then $$\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx=\text{ ?}, $$ where $\lfloor \space \rfloor$ is the "floor-function"
18
votes
2answers
292 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
5
votes
0answers
71 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
9
votes
3answers
213 views

How to evaluate improper integral $\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}dx$?

I'm trying to evaluate the improper integral, $$I(a)=\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}\mathrm{d}x,~~~\text{where }a\in\mathbb{R}^+.$$ Does this integral have a simple closed form ...
3
votes
1answer
71 views

Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
7
votes
2answers
131 views

Superelliptic Area Of $x^5+y^5=r^5$

$${\LARGE\int}_0^\tfrac\pi2\frac{dx}{\bigg(\sqrt[{\Large 5}]{\cos^5x+10\cos^3x\sin^2x+5\cos x\sin^4x}\bigg)^{\large 2}}~=~?$$ Its numerical value is about $1.40171345128228$. Maple, Mathematica, ...
13
votes
2answers
389 views

Evaluating $\int_0^{2\pi}\frac{dt}{\sqrt[4]{P(\cos t,\sin t)}}$

$${\LARGE\int}_0^{2\pi}\frac{dt}{\sqrt[{\LARGE 4}]{A\Big(\sin^8t+\cos^8t\Big)+B\Big(\sin^6t\cos^2t+\sin^2t\cos^6t\Big)+C~\sin^4t\cos^4t}}~=~?$$ where $A=0.3$, $B=-3.3$, and $C=10$. Its numerical ...
0
votes
2answers
89 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
11
votes
5answers
193 views

Do these integrals have a closed form? $I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$

The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign? $$I_1 = \int_{-\infty ...
3
votes
0answers
72 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
0
votes
0answers
52 views

Closed form definite integral. [closed]

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought $$\int_0^\infty \nu y^r e^{\nu y}\{1+\psi(e^{\nu y}-1)\}^{-\tfrac1\psi -1}dy.$$ where $r$ is any ...
9
votes
3answers
191 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
1
vote
2answers
98 views

Log Log Integrals

Evaluate the integrals \begin{align} I_{1} &= \int_{0}^{1} \ln^{2n}(x) \ \ln\left(\ln\left(\frac{1}{x}\right)\right) dx \end{align} and \begin{align} I_{2} = \int_{0}^{1} \ln^{2n}(x) \ ...
0
votes
1answer
70 views

Finding the area enclosed by Batman curve

I was reading this thread: Is this Batman equation for real? and I wondered if the area bounded by the curve had a nice closed form. The resulting integrals were slightly beyond me, so I thought I ...
2
votes
1answer
89 views

Integration of $\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$

Does the integral $$I(a)=\int_0^\pi\int_0^\pi[\sin x\sin y\ge a]\,dx\,dy$$ have any closed-form solution? This is the area under the contours of $\sin x\sin y$; $[\cdot]$ is the Iverson bracket, which ...
24
votes
2answers
586 views

Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi} \,\sqrt{\sin\phi+\cos\phi}}d\phi$$ Its approximate numeric value is ...
5
votes
1answer
101 views

Closed form of $\int_0^\infty \frac{\log(x)-\log(a)}{x-a}e^{-x} \mathrm{d}x$.

Can the integral $$ \int_0^\infty \frac{\log(x)-\log(a)}{x-a}e^{-x} \mathrm{d}x $$ be expressed in terms of some simple special function? I have searched through integral tables but couldn't find ...
11
votes
2answers
206 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
12
votes
1answer
191 views

Closed form for $\int_{-\infty}^0\operatorname{Ei}^3x\,dx$

Let $\operatorname{Ei}x$ denote the exponential integral: $$\operatorname{Ei}x=-\int_{-x}^\infty\frac{e^{-t}}tdt.\tag1$$ It's not difficult to find that ...
13
votes
2answers
726 views

Crazy $\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ {-x}\right)^2\frac{dx}{\sqrt x}$

Is there any chance to find a closed form for this integral? $$I=\int_0^\infty{_3F_2}\left(\begin{array}c\tfrac58,\tfrac58,\tfrac98\\\tfrac12,\tfrac{13}8\end{array}\middle|\ ...
13
votes
1answer
257 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
18
votes
2answers
394 views

Integral $\int_0^1\frac{\ln x}{x-1}\ln\left(1+\frac1{\ln^2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$ I \equiv \int_{0}^{1}{\ln\left(x\right) \over x - 1}\, \ln\left(1 + {1 \over \ln^{2}\left(x\right)}\right)\,{\rm d}x $$ Numerically, ...
8
votes
1answer
173 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
1
vote
0answers
61 views

Integral $\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx$

Does the following integral admit a closed-form expression? $$\int_0^{\infty } \frac{1}{(\alpha x^2 + 1) \left(- 2 \sqrt{\frac{ x^2}{x^2+1}}+2 x+\pi \right)} \, dx \;\; , \;\; 0 \leq \alpha \leq ...
25
votes
2answers
731 views

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
18
votes
4answers
492 views

Integral $\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx$

I have a trouble with this integral $$I=\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx.$$ Could you suggest how to evaluate it?
2
votes
0answers
65 views

Closed-form expression of a definite integral

Does this definite integral admit a closed-form in terms of elementary functions? $$\int_0^{\infty } \frac{x}{\left(x^4+1\right) \left(2 x^2-2 \arctan\left(x^2\right)+\pi \right)} \, dx.$$
10
votes
1answer
164 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
12
votes
1answer
292 views

How to prove $\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right)$?

How can I prove the following conjectured identity? $$\int_0^\pi\frac{\ln(2+\cos\phi)}{\sqrt{2+\cos\phi}}d\phi\stackrel?=\frac{\ln3}{\sqrt3}K\left(\sqrt{\frac23}\right),\tag1$$ where $K(x)$ is the ...
16
votes
4answers
556 views

Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$

I'm trying to find a closed form for the following integral: $$\mathcal{J}(n)=\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx\tag1$$ I have conjectured values ...
12
votes
2answers
142 views

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

I need to evaluate the following integral with a high precision: $$ I=\int_{0}^{\infty}\left[% {\pi^{2} \over 6} - {\rm Li}_2\left({\rm e}^{-x}\right) -{\rm Li}_2\left({\rm ...
1
vote
0answers
50 views

Closed form double integral $ \int_{a}^{c}dr \int_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{r_<^{\ell}}{r_>^{\ell+1}}$

Is there a closed form expression for $$ S_\ell = \int\limits_{a}^{c}dr \int\limits_{b}^{d} dr' \, \frac{r r'}{\sqrt{(r - a)(r' - b)(r-c)(r'-d)}} \frac{[\min( r , r')]^{\ell}}{[\max(r,r')]^{\ell+1}} ...