I’ve been looking at
It seems that it always evaluates in terms of sinX and π, where X is to be determined. For example:
So I guess there must be a closed form – the use of Γ(x)Γ(1−x) first comess to my mind because of the πxsinπx appearing. Note that the arguments are always the ratio of the exponents, like 14, 13 and 25. Is there any way of finding it? I’ll work on it and update with any ideas.
The integral reduces to finding
With a=n+1m which converges only if
Using series I find the solution is
I would like to make a supplementary calculation on BR's answer.
Let us first assume that 0<μ<ν so that the integral
converges absolutely. By the substitution x=tan2/νθ, we have
where the last equality follows from Euler reflexion formula.