\(4 \quad\) It is given that, for \(n \geqslant 0\),
\(I_{n}=\int_{0}^{1} x^{n} \mathrm{e}^{x^{3}} \mathrm{~d} x\)
(i) Show that \(I_{2}=\frac{1}{3}(\mathrm{e}-1)\).

(ii) Show that, for \(n \geqslant 3\),
\(3 I_{n}=\mathrm{e}-(n-2) I_{n-3} .\)

(iii) Hence find the exact value of \(I_{8}\).