Question Detail

This integral requires integration by parts, which may need to be applied twice. The formula for integration by parts is ∫ u dv = uv - ∫ v du.Let I = ∫ eˣ sin x dx.First application of integration by parts:Let u = sin x, dv = eˣ dxThen du = cos x dx, v = eˣSo, I = eˣ sin x - ∫ eˣ cos x dx (Equation 1)Second application of integration by parts (for ∫ eˣ cos x dx):Let u = cos x, dv = eˣ dxThen du = -sin x dx, v = eˣSo, ∫ eˣ cos x dx = eˣ cos x - ∫ eˣ (-sin x) dx = eˣ cos x + ∫ eˣ sin x dxNotice that ∫ eˣ sin x dx is our original integral I.Substitute this back into Equation 1:I = eˣ sin x - (eˣ cos x + I)I = eˣ sin x - eˣ cos x - INow, add I to both sides:2I = eˣ sin x - eˣ cos x2I = eˣ (sin x - cos x)Finally, divide by 2 and add the constant of integration C:I = (1/2) eˣ (sin x - cos x) + CThus, option C is correct.