Question

Evaluate ${\int }_{\pi /6}^{\pi /3}\frac{dx}{1+\sqrt{\tan x}}$

Answer 1

We have

$I=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{dx}{1+\sqrt{\tan x}}$

$=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{\sqrt{\tan x}}{1+\sqrt{\tan x}}dx$

$2I=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}dx={\left[x\right]}_{\frac{\pi }{6}}^{\frac{\pi }{3}}=\frac{\pi }{6}$

$\therefore I=\frac{\pi }{12}$

Hence, the required value of $I=\frac{\pi }{12}$