Question

The value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}}dx$ is

Answer 1

$I={\int }_0^{\pi /2}\frac{\sqrt{\tan x}}{\sqrt{\tan x}+\sqrt{\cot x}}dx$ __(i)

using $a+b-x$

$I={\int }_0^{\pi /2}\frac{\sqrt{\tan (\frac{\pi }{2}-x)}}{\sqrt{\tan (\frac{\pi }{2}-x)}+\sqrt{\cot (\frac{\pi }{2}-x)}}dx$

$={\int }_0^{\pi /2}\frac{\sqrt{\cot x}}{\sqrt{cotx}+\sqrt{\tan }}dx$

adding (i) and (ii)

$2I-{\int }_0^{\pi /2}\frac{\sqrt{\tan x}+\sqrt{\cot x}}{\sqrt{\tan x}+\cot x}dx\Rightarrow 2I={\int }_0^{\pi /2}dx=\frac{\pi }{2}$

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