4 It is given that, for \(n \geqslant 0\),
$$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x ^ { 3 } } \mathrm {~d} x$$
- Show that \(I _ { 2 } = \frac { 1 } { 3 } ( \mathrm { e } - 1 )\).
- Show that, for \(n \geqslant 3\),
$$3 I _ { n } = \mathrm { e } - ( n - 2 ) I _ { n - 3 }$$
- Hence find the exact value of \(I _ { 8 }\).