Evaluate the following integrals:

let $I=\int \frac{x^{2}}{x^{6}+a^{6}} d x$

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

Let $\mathrm{x}^{3}=\mathrm{t} \ldots \ldots$ (i)

$\Rightarrow 3 x^{2} d x=d t$

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

$I=\frac{1}{3 a^{3}} \tan ^{-1} \frac{t}{a^{3}}+c$

$\left[\right.$ since, $\left.\int \frac{1}{\mathrm{x}^{2}+(\mathrm{a})^{2}} \mathrm{dx}=\frac{1}{\mathrm{a}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{c}\right]$

$I=\frac{1}{3 a^{2}} \tan ^{-1} \frac{x^{2}}{a^{2}}+c$ [using (i)]