## Calculus 8th Edition

$$f^{\prime \prime\prime}(t)=12+\sin t$$ $\Longrightarrow$ $$f(x) =2 t^{3}+ \cos t +C t^{2}+D t+E$$ where $C,D$ and $E$ are arbitrary constants .
$$f^{\prime \prime\prime}(t)=12+\sin t$$ 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 .