Answer
The average value is
$$\overline{f}=\frac{1}{3}\arctan3.$$
Work Step by Step
We have to average function $f=\frac{1}{1+x^2}$ on the segment $[-3,3]$. The formula for average value of the function on a segment $[x_1,x_2]$ is
$$\overline{f}=\frac{\int_{x_1}^{x_2}f(x)dx}{x_2-x_1}.$$
Using the values given in this problem
$$\overline{f}=\frac{\int_{-3}^{3}\frac{1}{1+x^2}dx}{3-(-3)}=\frac{\left.\arctan x\right|_{-3}^3}{6}=\frac{1}{6}(\arctan 3-\arctan (-3))=\\\frac{1}{6}(\arctan 3+\arctan 3) = \frac{1}{3}\arctan 3.$$
