Evaluate ∫₀^(π/2) sin 2x log(tan x) dx.

Evaluate ∫₀^(π/2) sin 2x log(tan x) dx.

• A. 0
• B. 1
• C. π/2
• D. 2

Answer: A) 0

Explanation: Let I = ∫₀^(π/2) sin 2x log(tan x) dx. Put x = π/2 − t, sin 2x = sin(π−2t) = sin 2t, tan x = cot t, log(tan x) = −log(tan t). Integral becomes ∫₀^(π/2) sin 2t (−log(tan t)) dt = −I. So 2I = 0 → I = 0.