Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 50

$$\frac{2}{3}$$

\begin{align*} \int_{0}^{ \pi/2}\cos^3xdx&= \int_{0}^{ \pi/2}\cos^2x\cos xdx \\ &= \int_{0}^{ \pi/2}(1-\sin^2x)\cos xdx \\ &= \left(\sin x -\frac{1}{3}\sin ^3x\right) \bigg|_{0}^{\pi/2}\\ &=\frac{2}{3} \end{align*}