f ( x ) = sin ( 4 x ) , [ − π , π ] {\displaystyle f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]}
f a v g = 1 π − ( − π ) ∫ − π π sin ( 4 x ) d x = 1 2 π ∫ − π π sin ( 4 x ) d x {\displaystyle f_{avg}={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx}
u = 4 x d u = 4 d x 1 4 d u = d x {\displaystyle {\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 f(x)=\sin {(4x)}{\text{,}}\quad [-\pi ,\pi ]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0d26264e15aeb29ff049f70b6f38de6ed8709857)
![{\displaystyle {\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/2d7068b9cc23779b7a691ebbcdd78a156d8eab11)
