6.5 Average Value of a Function/2

Failed to parse (syntax error): {\displaystyle f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] \\[2ex] }

</math> \begin{align}

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{align} </math>

${\begin{aligned}u&=4x\\[2ex]du&=4dx\\[2ex]{\frac {1}{4}}du&=dx\end{aligned}}$

New upper limit: $4\pi =4(\pi )$
New lower limit: $-4\pi =4(-\pi )$