Question Detail

This integral requires integration by parts, usually twice.Let I = ∫ ex sin x dx.Use integration by parts: ∫ u dv = uv - ∫ v du.Choose u = sin x, dv = ex dx.Then du = cos x dx, v = ex.So, I = ex sin x - ∫ ex cos x dx.Now apply integration by parts again to ∫ ex cos x dx.Choose u = cos x, dv = ex dx.Then du = -sin x dx, v = ex.So, ∫ ex cos x dx = ex cos x - ∫ ex (-sin x) dx= ex cos x + ∫ ex sin x dx.Substitute this back into the expression for I:I = ex sin x - (ex cos x + ∫ ex sin x dx)I = ex sin x - ex cos x - INow, move -I to the left side:2I = ex sin x - ex cos xFactor out ex:2I = ex (sin x - cos x)Divide by 2 and add the constant of integration:I = (ex/2)(sin x - cos x) + C.This matches option C.