Failed to parse (syntax error): {\displaystyle f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] \\[2ex] }
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 = 1 8 π ∫ − 4 π 4 π sin ( u ) d u = − 1 8 π cos ( u ) | − 4 π 4 π = [ − 1 8 π cos ( 4 π ) ] − [ − 1 8 π cos ( − 4 π ) ] {\displaystyle {\begin{aligned}f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx\\[2ex]&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {(u)}{\frac {1}{4}}\,du={\frac {1}{8\pi }}\int _{-4\pi }^{4\pi }\sin(u)\,du\\[2ex]&=-{\frac {1}{8\pi }}\cos(u){\bigg |}_{-4\pi }^{4\pi }\\[2ex]&=\left[-{\frac {1}{8\pi }}\cos(4\pi )\right]-\left[-{\frac {1}{8\pi }}\cos(-4\pi )\right]\end{aligned}}}
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 {\begin{aligned}f_{avg}&={\frac {1}{\pi -(-\pi )}}\int _{-\pi }^{\pi }\sin {(4x)}\,dx={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\sin {(4x)}\,dx\\[2ex]&={\frac {1}{2\pi }}\int _{-4\pi }^{4\pi }\sin {(u)}{\frac {1}{4}}\,du={\frac {1}{8\pi }}\int _{-4\pi }^{4\pi }\sin(u)\,du\\[2ex]&=-{\frac {1}{8\pi }}\cos(u){\bigg |}_{-4\pi }^{4\pi }\\[2ex]&=\left[-{\frac {1}{8\pi }}\cos(4\pi )\right]-\left[-{\frac {1}{8\pi }}\cos(-4\pi )\right]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/99ef27570aff17433cf85e19fa704f34aeba38ce)
![{\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)
