Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 57

$$\frac{8}{3}$$

\begin{align*} \int_{-\pi/4}^{\pi/4} \sec^4 xdx&= \int_{-\pi/4}^{\pi/4} \sec^2 x\sec^2 xdx\\ &= \int_{-\pi/4}^{\pi/4} (1+\tan ^2 x)\sec^2 xdx\\ &=\tan x+\frac{1}{3}\tan^3x\bigg|_{-\pi/4}^{\pi/4}\\ & = \frac{8}{3} \end{align*}