Question

\[\int x^2 \sqrt{a^6 - x^6} \text{ dx}\]

Answer 1

\[\text{ Let I } = \int x^2 \sqrt{a^6 - x^6}\text{ \text{ dx}}\]
\[ = \int x^2 \sqrt{\left( a^3 \right)^2 - \left( x^3 \right)^2}\text{ dx}\]
\[Putting\ x^3 = t\]
\[ \Rightarrow 3 x^2 dx = dt\]
\[ \Rightarrow x^2 dx = \frac{dt}{3}\]
\[ \therefore I = \frac{1}{3}\int\sqrt{\left( a^3 \right)^2 - t^2}dt\]
\[ = \frac{1}{3} \left[ \frac{t}{2}\sqrt{\left( a^3 \right)^2 - t^2} + \frac{\left( a^3 \right)^2}{2} \text{ sin}^{- 1} \left( \frac{t}{a^3} \right) \right] + C\]
\[ = \frac{x^3}{6} \sqrt{a^6 - x^6} + \frac{a^6}{6} \text{ sin}^{- 1} \left( \frac{x^3}{a^3} \right) + C\]