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This integral requires integration by parts twice. Let I = ∫ ex sin x dx.First application of integration by parts: ∫ u dv = uv - ∫ v duLet u = sin x &implies; du = cos x dxLet dv = ex dx &implies; v = exSo, I = ex sin x - ∫ ex cos x dx.Second application of integration by parts (for ∫ ex cos x dx):Let u = cos x &implies; du = -sin x dxLet dv = ex dx &implies; v = exSo, ∫ 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, add I to both sides:2I = ex sin x - ex cos xFactor out ex and divide by 2:I = (ex/2)(sin x - cos x) + C.This matches option C.