Example 2.37. Find the average value of the function $u(x)=$ $=\sin ^{2}(2 x)$ on the half-interval $[0, \infty)$.
Solution.

$$
\begin{gathered}
M(u)=\lim _{x \rightarrow \infty}\left(\int_{0}^{x} \frac{\sin ^{2}(2 x) d x}{x}\right)=\lim _{x \rightarrow \infty} 0.5\left(\int_{0}^{x} \frac{1-\cos (4 x) d x}{x}\right)= \\
=\lim _{x \rightarrow \infty}\left(\frac{0.5 x-0.125 \sin 4 x}{x}\right)=0.5
\end{gathered}
$$