Prove that : ∫ log tan xdx ; x ∈ (0, π/2) = 0.

Question

Prove that : \(\int_0^{\pi / 2} \log \tan x d x=0.\)

Answer 1

\(\int_0^{\pi / 2} \log (\tan x) d x=0\)

L.H.S. \( I=\int_0^{\pi / 2} \log (\tan x) d x\)   .....(i)

\( I=\int_0^{\pi / 2} \log \left(\tan \left(\frac{\pi}{2}-x\right)\right) d x \quad\left[\tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta\right] \)

\(I=\int_0^{\pi / 2} \log (\cot x) d x\)  ......(ii)

\(\text { (i) }+ \text { (ii) }\)

\(2 I=\int_0^{\pi / 2}[\log (\tan x)+\log (\cot x)] d x \quad 2 I=\int_0^{\pi / 2}(\log \tan x \cdot \cot x) d x \)

\(2 I=\int_0^{\pi / 2} \log \mid d x \quad 2 I=\log \int_0^{\pi / 2} d x\)

\(2 I=0 \quad I=0=\text { R.H.S. Hence Proved. }\)