The value of $\int\limits^{\frac{\pi}{2}}_{0}\frac{\sqrt{cot\, x}}{\sqrt{tan\, x}+\sqrt{cot\, x}}dx$ is :

$\frac{\pi}{4}$

$I=\int\limits^{\frac{\pi}{2}}_{0}\frac{\sqrt{\cot\, x}}{\sqrt{\tan\, x}+\sqrt{\cot\, x}}dx$ ....(1)

$I=\int\limits^{\frac{\pi}{2}}_{0}\frac{\sqrt{\cot(\frac{\pi}{2}-x)}}{\sqrt{\tan(\frac{\pi}{2}-x)}+\sqrt{\cot(\frac{\pi}{2}-x)}}dx$

$I=\int\limits^{\frac{\pi}{2}}_{0}\frac{\sqrt{\tan x}}{\sqrt{\cot x}+\sqrt{\tan x}}dx$   ...(2)

eq. (1) + eq. (2)

$2I=\int\limits^{\frac{\pi}{2}}_{0}1dx$

so $2I=\frac{\pi}{2}$

so $I=\frac{\pi}{4}$