Question

Solve: $\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}\frac{\tan x}{\cot x+\tan x}dx$.

Answer 1

Let $I=$ $\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}\frac{\tan x}{\cot x+\tan x}dx$...…$\left(1\right)$

or, $I=$ $\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}\frac{\cot x}{\tan x+\cot x}dx$...…$\left(2\right)$

Now adding $\left(1\right)$ and $\left(2\right)$ we get,

$2I=$$\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}dx$

or, $2I=\frac{\pi }{2}$

or, $i=\frac{\pi }{4}$.