∫ cos²(x/2) + 2sin²(x/2) dx
= ∫ cos²(x/2) + sin²(x/2) + sin²(x/2) dx
= ∫ dx + (1/2)∫ 2sin²(x/2) dx
= ∫ dx + (1/2)∫ (1 - cosx) dx
= (3x - sinx)/2 + C