2024 HCI Promo Q10

It is given that \({{I}_{n}}=\displaystyle\displaystyle \displaystyle\int{\frac{{{x}^{n}}}{\sqrt{1-9{{x}^{4}}}}}\mathrm{d}x\).

1. Find \({{I}_{3}}\).[2]
2. By using the substitution \(u={{x}^{2}}\), find \({{I}_{1}}\).[4]
3. Using integration by parts and the result in part (a) , find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{1}{\sqrt{3}}}{x{{\sin }^{-1}}\left( 3{{x}^{2}} \right)}\,\,\mathrm{d}x\).[4]