int_0^(pi/4) (1 - tan x)/(1 + tan x) dx A. 0 B. 1 C. ln2 D. 1/2 ln2

GATE 2009 · Engineering Mathematics · Integration · medium

Answer: 1/2 ln2

1. Apply tangent subtraction identity: I = int_0^(pi/4) tan(pi/4 - x) dx
2. Substitute u = pi/4 - x: I = int_0^(pi/4) tan(u) du = [-ln|cos u|]_0^(pi/4)
3. Evaluate at bounds: I = -ln|cos(pi/4)| + ln|cos(0)| = -ln(1/sqrt(2)) + ln(1) = (1/2)ln2