Answer
$$
f(x) =-\frac{2x^6}{15}+\frac{x^3}{6}+\frac{x^8}{56}+Cx+\frac{x^2}{2}+D
$$
where $C,D$ are arbitrary constants .
Work Step by Step
$$
f^{\prime \prime}(x)= x^{6}-4x^{4}+ x+1
$$
The general anti-derivative of $
f^{\prime \prime}(x)=20 x^{3}-12 x^{2}+6 x
$ is
$$
f^{\prime}(x) =\frac{x^7}{7}-\frac{4x^5}{5}+\frac{x^2}{2}+x+ C
$$
Using the anti-differentiation rules once more, we find that
$$
f(x) =-\frac{2x^6}{15}+\frac{x^3}{6}+\frac{x^8}{56}+Cx+\frac{x^2}{2}+D
$$
where $C,D$ are arbitrary constants .
