Chapter 4 - Integration - 4.8 Average Value Of A Function And It's Applications - Exercises Set 4.8 - Page 335: 8

$$\frac{4}{\pi}$$

Using the average value equation: $f_{\text {ave}}=\frac{1}{-a+b} \int_{a}^{b} f(x) d x$ Thus: $f_{\text {ave}}=\frac{2}{\pi} \int_{-\pi / 4}^{\pi / 4} \sec ^{2} x d x=2 \cdot \frac{2}{\pi} =\frac{4}{\pi}$ We note: $f(x)=\sec ^{2} x$, $b=\frac{\pi}{4}$ and $a=-\frac{\pi}{4}$ $\tan x=F(x)$