Question Detail

To integrate ( \cos^3 x ), use the trigonometric identity ( \cos^2 x = 1 - \sin^2 x ).( \int \cos^3 x \,dx = \int \cos^2 x \cdot \cos x \,dx )( = \int (1 - \sin^2 x) \cos x \,dx )Now, use substitution. Let ( u = \sin x ).Then ( du = \cos x \,dx ).Substitute these into the integral:( = \int (1 - u^2) du )Integrate term by term:( = u - \frac{u^3}{3} + C )Substitute back ( u = \sin x ):( = \sin x - \frac{1}{3} \sin^3 x + C ).This matches option A.