Question

Evaluate: $\stackrel{\pi /2}{\underset{0}{\int }}\frac{dx}{1+\sqrt{\tan x}}$

Answer 1

${\int }_0^{\pi /2}\frac{dx}{1+\sqrt{\tan x}}=I$

$I={\int }_0^{\pi /2}\frac{dx}{1+\sqrt{\cot x}}$

$I+I={\int }_0^{\pi /2}(\frac{1}{1+\sqrt{\tan x}}+\frac{1}{1+\sqrt{\cot x}})dx$

$={\int }_0^{\pi /2}(\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}+\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}})dx$

$={\int }_0^{\pi /2}\left(1\right)dx$

$2I=\left(\frac{\pi }{2}\right)$

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