Explanation

(a)
Let $I = \int_{0}^{\pi/2}{\frac{\sqrt{\sin x}dx}{\sqrt{\cos x +}\sqrt{\sin x}}\ }$…(i)
$I = \int_{0}^{\pi/2}{\frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx}$ …(ii)
$\left\lbrack put\ \ x = \frac{\pi}{2} - x \right\rbrack$
On adding Eqs. (i) and (ii), we get
$2I = \int_{0}^{\pi/2}{dx\ }\ \ \Rightarrow \ I = \frac{\pi}{4}$