By definition, the average value of the function
equals to
For the first integral, we write
, and use the substitution
, for which
; we note that the endpoints
and
become
and
, respectively. So,
For the second integral, we use the trigonometric identity
to get
Therefore,
- The average value of the function
= ![{\displaystyle {\frac {2}{\pi }}{\bigg (}\int _{0}^{\pi /2}3\cos ^{3}x\,dx+\int _{0}^{\pi /2}2\cos ^{2}x\,dx{\bigg )}={\frac {2}{\pi }}{\bigg (}2+{\frac {\pi }{2}}{\bigg )}={\color {blue}{\frac {4}{\pi }}+1}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/0e0726b89ce9950661121fb500eb71367dc67767)