Show that $$ \int\_0^\pi \frac{x\,dx}{1+\cos^2(x)} = \frac{\pi^2}{2\sqrt{2}} $$
I tried using change of variable $x = \pi-y$ and then ended up with integral $\int\_0^\pi \frac{1}{1+\cos^2(y)}dy$ which I think doesn't make thing easier. I am wondering if there is any other clever change of variable or some trick to compute this original integral.