Answer
$$G(x)=-x^{-5}+2x^{-2}+2x +C$$
Work Step by Step
Given $$g(x) =\frac{5-4x^3+2x^6}{x^6}= 5x^{-6}-4x^{-3}+2$$
Then by using if $f(x)=x^{\alpha}\ \to\ \ F(x) = \frac{x^{\alpha+1}}{\alpha+1}+C$
\begin{align*}
G(x) &= \frac{5 }{-5}x^{-5}-\frac{4}{-2}x^{-2}+2x+C\\
&=-x^{-5}+2x^{-2}+2x +C
\end{align*}
To check
\begin{align*}
G'(x) &=5x^{-6}-4x^{-3}+2 \\
&=\frac{5-4x^3+2x^6}{x^6}\\
&=g(x)
\end{align*}