Calculating $\int_0^{\pi/2} \sqrt{\cot x} + \sqrt{\cos x} dx$ How should I solve the following integral:
$$\int_0^{\pi/2} (\sqrt{\cot x} + \sqrt{\cos x} )\,\mathrm dx$$

I think can use 
\begin{align}I&= \int_0^{\frac{\pi}{2}}\left(\sqrt{\cot (x)}+\sqrt{\cos (x)}\right)\,\mathrm dx\\&=\int _0^1\frac{1}{\cos(x)}\left(\sqrt{\cot (x)}+\sqrt{\cos (x)}\right)\, \mathrm d(\sin(x))\\&=\int _0^1\frac{1}{\sqrt{1-a^2}}\left(\sqrt{\frac{\sqrt{1-a^2}}{a}}+\sqrt{\sqrt{1-a^2}}\right)\,\mathrm da\\ &=\int _0^1 a^{-\frac{1}{2}}(1-a^2)^{-\frac{1}{4}}\,\mathrm da+\int _0^1 (1-a^2)^{-\frac{1}{4}}\,\mathrm da\\& =\frac{1}{2}\int _0^1 (a^2)^{-\frac{3}{4}}(1-a^2)^{-\frac{1}{4}}\,\mathrm d(a^2)+\frac{1}{2}\int _0^1 (a^2)^{-\frac{1}{2}}(1-a^2)^{-\frac{1}{4}}\,\mathrm d(a^2)\\ &=\frac{1}{2}B \left(\frac{1}{4}, \frac{3}{4}\right)+\frac{1}{2}B\left(\frac{1}{2},\frac{3}{4}\right)\end{align} with $B$ is beta function. Hope it can help.
