Question Detail

To evaluate the integral ∫ Cos³x dx, we use a trigonometric identity and substitution.First, rewrite Cos³x as Cos²x * Cos x.∫ Cos³x dx = ∫ Cos²x * Cos x dxNow, use the Pythagorean identity: Cos²x = 1 - Sin²x.∫ (1 - Sin²x) Cos x dxNow, perform a u-substitution:Let u = Sin x.Then, the differential du = (d/dx(Sin x)) dx = Cos x dx.Substitute u and du into the integral:∫ (1 - u²) duIntegrate term by term:∫ 1 du - ∫ u² du = u - (u³/3) + CFinally, substitute back u = Sin x:Sin x - (Sin³x)/3 + CThus, option A is correct.