Evaluate: `intx^2\ sqrt(a^6-x^6)\ dx`

Explanation

Let I `=int x^{2} sqrt{a^{6}-x^{6}} backslash` text{ `.{dx}}`

`=int x^{2} sqrt{(a^{3})^{2}-(x^{3})^{2}} {dx}`

Putting `x^{3}=t`

`Rightarrow 3 x^{2} d x=d t`

`Rightarrow x^{2} d x=frac{d t}{3}`

`therefore I=frac{1}{3} int sqrt{(a^{3})^{2}-t^{2}} d t`

`=frac{1}{3}[frac{t}{2} sqrt{(a^{3})^{2}-t^{2}}+frac{(a^{3})^{2}}{2} sin ^{-1}(frac{t}{a^{3}})]+C`

`=frac{x^{3}}{6} sqrt{a^{6}-x^{6}}+frac{a^{6}}{6} sin ^{-1}(frac{x^{3}}{a^{3}})+C`