Question

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

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $-\frac{\pi }{4}$

Answer 1

The correct option is C $\frac{\pi }{4}$

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

$\Rightarrow {\int }_0^{\pi /2}\frac{1}{1+\sqrt{\tan (\pi /2-x)}}dx$

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

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

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

$I+I={\int }_0^{\pi /2}1dx$

$\Rightarrow [x{]}_0^{\pi /2}=2I$

$2I=\pi /2$

$\Rightarrow I=\pi /4$.