Integrate the function: $\frac{x^{2}}{\sqrt{x^{6}+a^{6}}}$

(A) Let $x^{3} = t$.
Then,differentiating both sides with respect to $x$,we get $3x^{2} dx = dt$,which implies $x^{2} dx = \frac{1}{3} dt$.
Substituting these into the integral:
$\int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} dx = \int \frac{1}{\sqrt{(x^{3})^{2} + (a^{3})^{2}}} \cdot \frac{1}{3} dt$
$= \frac{1}{3} \int \frac{dt}{\sqrt{t^{2} + (a^{3})^{2}}}$
Using the standard integral formula $\int \frac{dx}{\sqrt{x^{2} + a^{2}}} = \log |x + \sqrt{x^{2} + a^{2}}| + C$,we get:
$= \frac{1}{3} \log |t + \sqrt{t^{2} + (a^{3})^{2}}| + C$
Substituting $t = x^{3}$ back into the expression:
$= \frac{1}{3} \log |x^{3} + \sqrt{x^{6} + a^{6}}| + C$,where $C$ is an arbitrary constant.