Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 1

$$\sin x-\frac{1}{3} \sin ^{3} x+c $$

\begin{aligned} \int \cos ^{3} x d x &=\int \cos ^{2} x \cos x d x \\ &=\int\left(1-\sin ^{2} x\right) \cos x d x \\ &=\int(\cos x d x)-\int\left(\sin ^{2} x\right) \cos x d x \\ &=\sin x-\frac{1}{3} \sin ^{3} x+c \end{aligned}