Step-by-Step Solution
Key Concept: Use the property ∫_0^a f(x) dx = ∫_0^a f(a-x) dx.
Step 1: Let I = ∫_0^π (x sin x)/(1 + cos^2 x) dx. Step 2: I = ∫_0^π ((π-x) sin x)/(1 + cos^2 x) dx = π ∫_0^π sin x/(1 + cos^2 x) dx - I. Step 3: 2I = π ∫_0^π sin x/(1 + cos^2 x) dx. Step 4: Substitute cos x = t, -sin x dx = dt, limits 1 to -1: ∫_0^π sin x/(1 + cos^2 x) dx = ∫_{-1}^1 1/(1+t^2) dt = [tan^{-1} t]_{-1}^1 = π/2. Step 5: 2I = π*(π/2) = π^2/2, I = π^2/4.
Correct Answer: A