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
52 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
97 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 ...
5
votes
3answers
165 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\\ ...
5
votes
1answer
101 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: ...
5
votes
0answers
62 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 ...
8
votes
3answers
192 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
69 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 ...
6
votes
2answers
113 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
356 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 ...
8
votes
3answers
127 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 ...
9
votes
3answers
177 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.
4
votes
1answer
84 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
2
votes
1answer
85 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 ...
22
votes
2answers
557 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 ...
9
votes
2answers
195 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
182 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
638 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
245 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 ...
17
votes
2answers
324 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
167 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 ...
25
votes
2answers
625 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
433 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
198 views

Integrating a complicated function

After spending a couple of weeks, I was able to find the solution to a certain differential equation, given below (Well they are the eigenfunctions to be exact): $$y_n(x) = ...
0
votes
3answers
604 views

Integral $\int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx$

I recently got stuck on evaluating the following integral, $$ \int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx. $$ Is it possible to evaluate this integral in a closed form? I am not sure if there is ...
7
votes
1answer
198 views

Evaluating $\int_0^{\frac{1}{2}}\log^2(2\sin(\pi x))\cos(\pi(1-2x))dx$ [closed]

How can we prove following formulas $$\int_0^{\frac{1}{2}}\log^2(2\sin(\pi x))\cos(\pi(1-2x))dx=\frac{-1}{4}$$ or $$\int_0^{\frac{1}{2}}\log^3(2\sin(\pi x))\cos(\pi(1-2x))dx=\frac{\pi^2+6}{16}$$
11
votes
5answers
457 views

Closed form for $\int_{0}^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right)\,{\rm d}x$

How can I find a closed form for the following integral $$ \int_0^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right) \,{\rm d}x $$
13
votes
2answers
267 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
11
votes
1answer
177 views

Integral $\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx$

Consider the following integral: $$\mathcal{A}=\int_0^\infty\exp\left(-\sqrt2\,x^2\right)\,\operatorname{erfi}(x)\,\log(x)\,x^3\,dx,\tag1$$ where $\operatorname{erfi}(x)$ denotes the imaginary error ...
18
votes
3answers
389 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 ...
19
votes
3answers
539 views

Integral $\int_0^\infty x^2\,e^{-x^2}\operatorname{erf}(x)\,\log(x)\,dx$

I need to evaluate this integral: $$I=\int_0^\infty x^2\,e^{-x^2}\operatorname{erf}(x)\,\log(x)\,dx\tag1$$ I tried to do this in Mathematica and it returned a result of the form ...
23
votes
2answers
262 views

Need help with $\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx$

I need help with this integral: $$\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx\approx0.20597312051214...$$ Is it possible to evaluated it in a closed form?
10
votes
2answers
337 views

Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $

Is there a closed form for $|r|<1$ and $\ell>0$ integer? The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available. Integrating ...
1
vote
3answers
205 views

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
17
votes
1answer
452 views

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate ...
21
votes
2answers
631 views

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

Could you suggest any ideas how to evaluate this integral? Is there a closed-form result? $$\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$$
21
votes
3answers
370 views

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in ...
19
votes
2answers
320 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
7
votes
1answer
191 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
22
votes
2answers
614 views

Closed form for $\int_0^\infty\frac{\sin x\,\cdot\,\operatorname{Ci}x-\cos x\,\cdot\,\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx$

Is it possible to find a closed form for this integral? $$\mathcal{S}=\int_0^\infty\frac{\sin x\cdot\operatorname{Ci}x-\cos x\cdot\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx,$$ where $\operatorname{Ci}x$ ...
5
votes
1answer
184 views

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

How can I find a closed form for $$\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx, n\in\mathbb{N}$$
33
votes
4answers
1k views

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

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx.$$ I do not have a strong reason to be sure it exists, but I would be ...
27
votes
1answer
729 views

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

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) \, dx = 4 \pi ...
26
votes
2answers
1k views

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

This is somewhat similar to my previous question: Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$ Is it possible to find a closed form ...
8
votes
0answers
113 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) ...
25
votes
2answers
371 views

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
27
votes
2answers
861 views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)dx$$
21
votes
1answer
261 views

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

I need to evaluate the following integral: $$\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx.$$ Could you suggest how to find a closed form for it? I am not sure if there is ...
24
votes
3answers
520 views

Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$

There is a curious known integral: $$\int_0^1\frac{\ln\left(1+x^{2+\sqrt{3\vphantom{\large3}}}\right)}{1+x}dx=\frac{\pi^2}{12}\left(1-\sqrt{3\vphantom{\large3}}\right)+\ln ...
10
votes
1answer
259 views

What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...