Question Detail

This integral requires integration by parts, which is given by ∫ u dv = uv - ∫ v du.Let ∫ I = ex sinx dx.Choose u = sin x and dv = ex dx.Then du = cos x dx and v = ex.Substitute into the formula:I = ex sin x - ∫ ex cos x dx.Now, we need to evaluate ∫ ex cos x dx. We apply integration by parts again.Let u' = cos x and dv' = ex dx.Then du' = -sin x dx and 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 - I.Add I to both sides:2I = ex sin x - ex cos x.Finally, solve for I:I = (ex sin x - ex cos x)/2 + CI = ex/2 (sin x - cos x) + C. This matches option C.