Question

\[\int\limits_0^{\pi/2} \log \tan x\ dx .\]

Answer 1

\[\text{Let, }I = \int_0^\frac{\pi}{2} \log \tan x\ dx ...................(1)\]

\[ = \int_0^\frac{\pi}{2} \log \tan\left( \frac{\pi}{2} - x \right) dx ....................\left[ Using, \int_0^a f\left( x \right) dx = \int_0^a f\left( a - x \right) dx \right]\]

\[ = \int_0^\frac{\pi}{2} \log cot x\ dx ....................(2)\]

\[\text{Adding (1) and (2) we get}\]

\[2I = \int_0^\frac{\pi}{2} \log \tan x d x + \int_0^\frac{\pi}{2} \log cotx\ dx\]

\[ = \int_0^\frac{\pi}{2} \log\left( \tan x \times cotx \right)dx\]

\[ = \int_0^\frac{\pi}{2} \log1 dx = 0\]

\[\text{Hence, }I = 0\]