Question

Find the value of ∫(sin(n + (1/2)) x/sin (x/2)) dx for x ∈ [0,π].

Answer 1

Let

Now, I_n+1 – I_n

Thus, I_n+1 – I_n 

I_n+1 = I_n = I_{n –1} = I_{n –2} = ... = I₁ = I₀

Therefore,

Hence, I_n = π, n ≥ 0.