Answer
$$
f^{\prime \prime\prime}(t)=12+\sin t
$$
$\Longrightarrow $
$$
f(x) =2 t^{3}+ \cos t +C t^{2}+D t+E
$$
where $C,D $ and $E$ are arbitrary constants .
Work Step by Step
$$
f^{\prime \prime\prime}(t)=12+\sin t
$$
The general anti-derivative of $
f^{\prime \prime\prime}(t)=12+\sin t
$ is
$$
f^{\prime\prime}(x) =12t -\cos t +C_{1} \\
$$
Using the anti-differentiation rules once more, we find that
$$
f^{\prime}(x) =6t^{2}- \sin t +C_{1} t+D
$$
Using the anti-differentiation rules once more, we find that
$$
f(x) =2 t^{3}+ \cos t +C t^{2}+D t+E
$$
where $C=\frac{1}{2}C_{1},D, E$ are arbitrary constants .
