If $\int \sqrt{\frac{x^4}{a^6+x^6}} d x=g(x)+C$, then $g(x)=$

$\frac{1}{3} \log \left|x^3+\sqrt{a^6+x^6}\right|$

We have,

$I =\int \sqrt{\frac{x^4}{a^6+x^6}} d x$

$\Rightarrow I =\int \frac{x^2}{\sqrt{\left(a^3\right)^2+\left(x^3\right)^2}} d x$

$\Rightarrow I =\frac{1}{3} \int \frac{1}{\sqrt{\left(a^3\right)^2+\left(x^3\right)^2}} d\left(x^3\right)$

$\Rightarrow I=\frac{1}{3} \log \left|x^3+\sqrt{x^6+a^6}\right|+C$

∴  $g(x)=\frac{1}{3} \log \left|x^3+\sqrt{a^6+x^6}\right|$