f ( x ) = sin ( 4 x ) , [ − π , π ] f a v g = 1 π − ( − π ) ∫ − π π sin ( 4 x ) d x = 1 2 π ∫ − π π sin ( 4 x ) d x = 1 2 π ∫ − 4 π 4 π sin u 1 4 d u < m a t h > u = 4 x d u = 4 d x 1 4 d u = d x {\displaystyle {\begin{aligned}f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {u}{\frac {1}{4}}\,du\end{aligned}}<math>{\begin{aligned}u&=4x\\[2ex]du&=4dx\\[2ex]{\frac {1}{4}}du&=dx\end{aligned}}}
New upper limit: 4 π = 4 ( π ) {\displaystyle 4\pi =4(\pi )} New lower limit: − 4 π = 4 ( − π ) {\displaystyle -4\pi =4(-\pi )}
![{\displaystyle {\begin{aligned}f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {u}{\frac {1}{4}}\,du\end{aligned}}<math>{\begin{aligned}u&=4x\\[2ex]du&=4dx\\[2ex]{\frac {1}{4}}du&=dx\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1dd6636fa64f7deb3edb4913f9fb3804abfabfdf)
