Question

Find the value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left(\tan x\right)dx$.

Answer 1

Let $I=$ $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left(\tan x\right)dx$........(1).

or, $I=$ $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log (\tan \frac{\pi }{2}-x)dx$

or, $I=$$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log \left(\cot x\right)dx$........(2).

Now adding (1) and (2) we get,

$2I=$ $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\log (\tan x.\cot x)dx$

or, $2I=0$

or, $I=0$.