Example 2.66. The velocity $v$ of a point moving in a straight line changes with time $t$ according to the law $v(t)=$ $=t \sin 2 t$. Determine the path $s$ traveled by the point from the start of the motion to the moment of time $t=\frac{\pi}{4}$ units of time.
Solution. Since the path $d s=v(t) d t$, then

$$
\begin{aligned}
s=\int_{t_{\text {nav }}}^{t_{\text {toon }}} v(t) d t & =\int_{0}^{\pi / 4} t \sin 2 t d t=\left|\begin{array}{l}
u=t, d u=d t, d v=\sin 2 t d t \\
v=\int \sin 2 t \frac{d(2 t)}{2}=-0.5 \cos 2 t
\end{array}\right|= \\
& =-0.5 t \cos 2 t\left|_{0}^{\pi / 4}\right.+0.5 \int_{0}^{\pi / 4} \cos 2 t d t=
\end{aligned}
$$

$=-0.5 t \cos 2 t\left|_{0}^{\pi / 4}\right.+0.25 \sin 2 t\left|_{0}^{\pi / 4}\right.=-\frac{\pi}{8} \cos \frac{\pi}{2}+0.25 \sin \frac{\pi}{2}=0.25$.

Note that if the formula $v(t)$ changed sign on the interval $[0, \pi / 4]$, then we would need to find the absolute value of the integral and then sum all the results:

$$
s(t)=\sum_{k=1}^{n-1}\left|\int_{t_{k}}^{t_{k+1}} v(t) d t\right|
$$