All Questions
5
questions
11
votes
0
answers
255
views
Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$
I tried to solve this integral and got it, I showed firstly
$$\int_0^1 \frac{\ln^2(x+1)+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx=2\Im\left[\text{Li}_3(1+i) \right] $$
and for other integral
$$\int_0^...
- 1,637
5
votes
2
answers
368
views
Closed form of the sum $s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}$
I am interested to know if the following sum has a closed form
$$s_4 = \sum_{n=1}^{\infty}(-1)^n \frac{H_{n}}{(2n+1)^4}\tag{1}$$
I stumbled on this question while studying a very useful book about ...
- 12.7k
2
votes
5
answers
485
views
Integral involving Dilogarithm $\int_{1/2}^{1} \frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$
I need your help in evaluating the following integral in closed form. $$\displaystyle\int\limits_{0.5}^{1}
\frac{\mathrm{Li}_{2}\left(x\right)\ln\left(2x - 1\right)}{x}\,\mathrm{d}x$$
Since the ...
- 407
31
votes
3
answers
2k
views
What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?
Some time ago I asked How to find $\displaystyle{\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$.
Thanks to great effort of several MSE users, we now know that
\begin{align}
\int_0^1\frac{\ln^3(1+x)\,\ln ...
- 5,342
33
votes
4
answers
3k
views
Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$
This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here:
Prove:
$$\int_0^1\ln(1-x)\ln(1+x)\ln^...
- 5,342