Non-Elementary Integration · Challenge problem

Let $A=\displaystyle\int_0^{\pi/2}\sqrt{\sin x}\,dx$ and $B=\displaystyle\int_0^{\pi/2}\dfrac{dx}{\sqrt{\sin x}}$; neither has an elementary value. Using $\displaystyle\int_0^{\pi/2}\sin^{2p-1}x\cos^{2q-1}x\,dx=\dfrac{\Gamma(p)\Gamma(q)}{2\,\Gamma(p+q)}$, $\Gamma\!\left(\tfrac12\right)=\sqrt\pi$ and $\Gamma(s+1)=s\,\Gamma(s)$, find $AB$.