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Return to GATE2009 q25.
$\int^{\pi/4}_0 (1-tanx)/(1+tanx)\,dx $
(A) 0
(B) 1
(C) ln 2
(D) 1/2 ln 2
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.
GATECSE