View source for GATE2009 q25 ← GATE2009 q25

$\int^{\pi/4}_0 (1-tanx)/(1+tanx)\,dx $

(A) 0

(B) 1

(C) ln 2

(D) 1/2ln 2

Solution by Happy Mittal[edit]

We know that $(1-\tan x)/(1+\tan x) = \tan(\pi;/4 - x)$, so $$\int^{\pi/4}_0 (1-tanx)/(1+tanx)\,dx = \int^{\pi/4}_0 tan(π/4 - x)\,dx = -\left[\ln \sec(\pi/4 - x)\right]^{\pi/4}_0 = 1/2 \ln 2$$ So option (D) is correct.