# What is the average value of the function f(x) = sec x tan x on the interval [0,pi/4]?

Mar 23, 2016

It is $\frac{4 \left(\sqrt{2} - 1\right)}{\pi}$

#### Explanation:

The average value of a function $f$ on an interval $\left[a , b\right]$ is

$\frac{1}{b - a} {\int}_{a}^{b} f \left(x\right) \mathrm{dx}$

So the value we seek is

$\frac{1}{\frac{\pi}{4} - 0} {\int}_{0}^{\frac{\pi}{4}} \sec x \tan x \mathrm{dx}$

$= \frac{4}{\pi} {\left[\sec x\right]}_{0}^{\frac{\pi}{4}}$

$= \frac{4}{\pi} \left[\sec \left(\frac{\pi}{4}\right) - \sec \left(0\right)\right]$

$= \frac{4}{\pi} \left[\sqrt{2} - 1\right]$

$= \frac{4 \left(\sqrt{2} - 1\right)}{\pi}$