All Questions
Tagged with closed-form calculus
307
questions
595
votes
14
answers
384k
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 ...
- 13.2k
71
votes
5
answers
4k
views
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
Show that :
$$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\...
- 3,955
11
votes
2
answers
689
views
Infinite Series $\sum\limits_{k=1}^{\infty}\frac{k^n}{k!}$
How can I find the value of the sum $\sum_{k=1}^{\infty}\frac{k^n}{k!}$?
for example for $n=6$, we have
$$\sum_{k=1}^{\infty}\frac{k^6}{k!}=203e.$$
- 5,626
37
votes
4
answers
2k
views
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
Find the closed form of $$\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}.$$
We can use the Fourier series of $e^{-bx}$ ($|x|<\pi$) to find $$\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}.$$ But here ...
- 45.3k
66
votes
2
answers
4k
views
Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
I've found here the following integral.
$$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}$$
I've never seen it before and I also didn'...
- 12.4k
18
votes
5
answers
3k
views
Infinite Series $\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}$
I'm looking for a way to prove
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$
I know that
$$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{1}{4^{2m+...
user97601
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 ...
- 12.4k
46
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 $...
- 47.3k
24
votes
4
answers
5k
views
How to find the integral $\int_{0}^{\infty}\exp(- (ax+b/x))\,dx$?
How do I find
$$\large\int_{0}^{\infty}e^{-\left(ax+\frac{b}{x}\right)}dx$$
where $a$ and $b$ are positive numbers?
This is not a homework question. I will be quite happy if somebody can come up ...
- 8,937
44
votes
2
answers
3k
views
What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?
I have been trying to evaluate the series $$\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right) = 1.133928715547935...$$ using integration techniques, and I was wondering if ...
- 2,017
36
votes
6
answers
2k
views
Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$
One of the possible ways of computing the series is to obtain the generating function, but
this might be a tedious, hard work, pretty hard to obtain. What would you propose then?
$$\sum_{n=1}^{\...
- 44.4k
54
votes
2
answers
4k
views
Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$
Is it possible to evaluate this integral in a closed form?
$$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\...
- 2,793
43
votes
6
answers
2k
views
Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $ [duplicate]
I know that $$ \int_0^\infty \frac{\log x}{\exp x} = -\gamma $$ where $ \gamma $ is the Euler-Mascheroni constant, but I have no idea how to prove this.
The series definition of $ \gamma $ leads me ...
- 2,760
32
votes
2
answers
2k
views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
- 1,475
22
votes
8
answers
2k
views
Improper Integral $\int\limits_0^1\frac{\ln(x)}{x^2-1}\,dx$
How can I prove that?
$$\int_0^1\frac{\ln(x)}{x^2-1}\,dx=\frac{\pi^2}{8}$$
I know that
$$\int_0^1\frac{\ln(x)}{x^2-1}\,dx=\sum_{n=0}^{\infty}\int_0^1-x^{2n}\ln(x)\,dx=\sum_{n=0}^{\infty}\frac{1}{(...
user97601