Question

Prove that ∫(sin nx/sin x) dx for x ∈ [0,π] = {(0, when n is even), (π, when n is odd)

Answer 1

sin nx – sin(n – 2)x = 2 cos(n – 1)x sin x

I_n = 0 + I_n–2 = I_n–4 = ... = I₂ or I₁

according as n is even or odd.

Case I: When n is even

Case II: When n is odd