Question

The value of ${\int }_0^{\pi /2}\frac{\sqrt{\cot x}}{\sqrt{\left(\cot x\right)}+\sqrt{\left(\tan x\right)}}$dx is

• A. $\pi /4$
• B. $\pi /2$
• C. $\pi$
• D. None of these.

Answer 1

The correct option is A $\pi /4$

Let $I={\int }_0^{\pi /2}\frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}}dx$ ...(1)
Using integration property
$I={\int }_0^{\pi /2}\frac{\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}}dx$ ...(2)
Adding (1) and (2)
$2I={\int }_0^{\pi /2}dx=\frac{\pi }{2}\Rightarrow I=\frac{\pi }{4}$