Solution

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Example Solution:
The acceleration of the body defined by

$$a(t) = bt$$

We know that acceleration is the time derivative of velocity:

$$a(t) = \frac{d v(t)}{d t}$$

To find the velocity $v(t)$, we integrate $a(t)$ with respect to time:

$$v(t) = \int a(t) \, dt = \int b t \, dt$$

If the initial velocity is $v(0) = 0$, then the velocity becomes:

$$v(t) = \frac{b t^2}{2}$$

Likewise, integrate $v(t)$ with respect to time:

$$x(t)= \int v(t) \, dt = \frac{b}{2} \int t^2 \, dt$$

From where the coordinate from time, considering the initial conditions:

$$\boxed{x(t)=\frac{bt^3}{6}}$$