I need an asymptotic expansion of J(n)
$J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$
Can anybody help to find the asymptotic analytically or at least via numirical calculation please? Also, I wonder if there is a graphical interpretation of the result exists? Best to my knowledge the integral is not very simple to get an answer. Thank you for any help.