Question Detail

To evaluate this integral, we can rewrite $\cos^3 x$ using the Pythagorean 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 u-substitution.Let $u = \sin x$.Then $du = \frac{d}{dx}(\sin x) dx = \cos x dx$.Substitute $u$ and $du$ into the integral:$\int (1 - u^2) du$Integrate with respect to $u$:$u - \frac{u^3}{3} + C$Finally, substitute $\sin x$ back for $u$:$\sin x - \frac{1}{3} \sin^3 x + C$This matches option A.