Answer
$$
f^{\prime \prime \prime}(t)=\sqrt{t}-2 \cos t
$$
The required function is
$$
f(x) =\frac{8}{105}t^{\frac{7}{2}} -2 \sin t +C t^{2}+D t+E
$$
where $C,D, E$ are arbitrary constants .
Work Step by Step
$$
f^{\prime \prime \prime}(t)=\sqrt{t}-2 \cos t
$$
The general anti-derivative of $
f^{\prime \prime \prime}(t)=\sqrt{t}-2 \cos t
$ is
$$
f^{\prime\prime}(x) =\frac{2}{3}t^{\frac{3}{2}} -2 \sin t +C_{1} \\
$$
Using the anti-differentiation rules once more, we find that
$$
f^{\prime}(x) =\frac{4}{15}t^{\frac{5}{2}} +2 \cos t +C_{1} t+D
$$
Using the anti-differentiation rules once more, we find that
$$
f(x) =\frac{8}{105}t^{\frac{7}{2}} -2 \sin t +C t^{2}+D t+E
$$
where $C=\frac{1}{2}C_{1},D, E$ are arbitrary constants .
