All Questions
4
questions
11
votes
0
answers
436
views
Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?
Using
$$\text{Li}_{2a+1}(x)-\text{Li}_{2a+1}(1/x)=\frac{i\,\pi\ln^{2a}(x)}{(2a)!}+2\sum_{k=0}^a \frac{\zeta(2a-2k)}{(2k+1)!}\ln^{2k+1}(x)\tag{1}$$
and
$$\int_0^1x^{n-1}\operatorname{Li}_a(x)\mathrm{d}...
35
votes
0
answers
2k
views
Are these generalizations known in the literature?
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(...
6
votes
1
answer
494
views
Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?
We know how hard these integrals
$$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx;
\int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx;
...$$
can be. So I decided to come up with a ...
13
votes
2
answers
522
views
On the integral $\int_{0}^{1/2}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz$
This questions is related to my previous one.
I am interested in a explicit evaluation in terms of Euler sums for
$$ \int_{0}^{\pi/4}\text{Li}_3(\cos^2\theta)\,d\theta = \frac{1}{2}\int_{0}^{1/2}\...