To calculate the instantaneous velocity from a given acceleration, the differentiation must be reversed and integrated over the acceleration. For the velocity, in the case of uniformly accelerated motion ($\vec{a}=\mathrm{const}$), the indefinite integral is then obtained:
\begin{equation}
\vec{v} = \int \vec{a}\,\mathrm{d}t = \vec{a}t + \vec{v}_0
\end{equation}
The constant of integration is then equal to the initial velocity of the accelerating body. To obtain the instantaneous position from the velocity, a further integration with respect to time must be performed. Taking the integration rules into account, this ultimately leads to the general distance-time law for uniformly accelerating bodies:
\begin{equation}
\vec{r}(t) = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{r}_0
\end{equation}
Further integration yields a new integration constant $\vec{r}_0$, which is now interpreted as the initial position of the body relative to the origin. The quadratic relationship between distance $s$ and time $t$ is particularly noteworthy. Depending on whether the acceleration is constant ($a=\mathrm{const}$), the velocity ($v=\mathrm{const}$), or the position ($s=\mathrm{const}$), the temporal functions for $a$, $v$, and $s$ look different. The general distance-time law finds an important application, for example, in mobile phone navigation software. If the GPS signal fails while driving through a tunnel, for example, the software can temporarily compensate for this using the acceleration sensors built into modern smartphones.