Answer
$$\overline f = \frac{\pi }{4}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{1}{{{x^2} + 1}}{\text{ on the interval }}\left[ { - 1,1} \right] \cr
& {\text{Find the average value using }}\overline f = \frac{1}{{b - a}}\int_a^b {f\left( x \right)} dx \cr
& \overline f = \frac{1}{{1 - \left( { - 1} \right)}}\int_{ - 1}^1 {\frac{1}{{{x^2} + 1}}} dx \cr
& \overline f = \frac{1}{2}\left[ {{{\tan }^{ - 1}}x} \right]_{ - 1}^1 \cr
& {\text{Integrate}} \cr
& \overline f = \frac{1}{2}\left[ {{{\tan }^{ - 1}}\left( 1 \right) - {{\tan }^{ - 1}}\left( { - 1} \right)} \right] \cr
& \overline f = \frac{1}{2}\left[ {\frac{\pi }{4} + \frac{\pi }{4}} \right] \cr
& \overline f = \frac{1}{2}\left( {\frac{\pi }{2}} \right) \cr
& \overline f = \frac{\pi }{4} \cr
& \cr
& {\text{Graph}} \cr} $$