TITLE: Compute the integral $\int_0^{\frac\pi2}\frac{\mathrm d \theta}{\sqrt{\sin \theta}}$
QUESTION [5 upvotes]: Compute the Riemann integral
$$\int_0^{\frac\pi2}\frac{\mathrm d \theta}{\sqrt{\sin \theta}}$$
It seems very difficult, I don't know how to go ahead.
Thank you very much for your help!

REPLY [3 votes]: In general we have 
$$2\int^{\pi/2}_0\sin^{2x-1}(\theta)\cos^{2y-1}(\theta)\,d\theta=B(x,y)$$