STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

and $I = \int_0^{\pi /2} {\frac{{\sqrt {\cos \left( {\frac{\pi }{2} - x} \right)} }}{{\sqrt {\sin \left( {\frac{\pi }{2} - x} \right)} + \sqrt {\cos \left( {\frac{\pi }{2} - x} \right)} }}dx} $

$I = \int_0^{\pi /2} {\frac{{\sqrt {\sin x} }}{{\sqrt {\cos x + } \sqrt {\sin x} }}} \,dx$....$(ii)$

Adding $(i)$ and $(ii),$ we get

$2I = \int_0^{\pi /2} {(1)dx = \frac{\pi }{2} \Rightarrow I = \frac{\pi }{4}} $.