Time derivative of position function $x(t)$ is velocity function $v(t)$,
$$\tag{1} \frac{d}{dt} x(t) = v(t), $$
and integral of velocity function $v(t)$ with respect to time $t$ will give position function $x(t)$,
$$\tag{2} x(t) = \int v(t) dt + C, $$
with $C$ is constant of integration. If it becomes definite integral
$$\tag{3} x(t) - x(t_0) = \int_{t_0}^t v(t) dt, $$
then the integration constant $C$ has the role as initial position $x(t_0)$.
- William Moebs, Samuel J. Ling, Jeff Sanny, “Finding Velocity and Displacement from Acceleration”, in University Physics Volume 1, OpenStax, Houston, Texas, 19 Sep 2016, url https://openstax.org/books/university-physics-volume-1/pages/3-6-finding-velocity-and-displacement-from-acceleration [20221124].