Correct option is (c) 0
Let,
\(I = \int \limits_0^\frac \pi 2\log(\tan x)dx\)
Using the property
\(\int \limits_a ^b f(x)dx = \int \limits_a ^bf(a + b- x)dx,\) we get:
\(I = \int \limits _0^\frac \pi 2\log(\tan (\frac \pi 2 - x))dx\)
⇒ \(I = \int\limits_0^\frac \pi 2 \log(\cot x)dx\)
\(= \int \limits_0^\frac \pi 2 \log\frac 1{\tan x}dx\)
\(= - \int \limits_0^\frac \pi 2\log(\tan x)dx\)
\(I = -I\)
⇒ \(I = 0\)
