7.2 definite integral — Maths STD 12 Science — Question

$ = \int_0^{\pi /2} {\frac{{\sqrt {\cot \left( {\frac{\pi }{2} - x} \right)} }}{{\sqrt {\cot \left( {\frac{\pi }{2} - x} \right)} + \sqrt {\tan \left( {\frac{\pi }{2} - x} \right)} }}dx} $

$ = \int_0^{\pi /2} {\frac{{\sqrt {\tan x} }}{{\sqrt {\tan x} + \sqrt {\cot x} }}dx} $.....$(ii)$

Now adding $(i)$ and $(ii),$ we get

$2I = \int_0^{\pi /2} {\frac{{\sqrt {\cot x} + \sqrt {\tan x} }}{{\sqrt {\tan x} + \sqrt {\cot x} }}dx = [x]_0^{\pi /2} \Rightarrow I = \frac{\pi }{4}} $.