8.
$$I _ { n } = \int _ { 0 } ^ { 2 } ( x - 2 ) ^ { n } \mathrm { e } ^ { 4 x } \mathrm {~d} x \quad n \geqslant 0$$
- Prove that for \(n \geqslant 1\)
$$I _ { n } = - a ^ { n - 2 } - \frac { n } { 4 } I _ { n - 1 }$$
where \(a\) is a constant to be determined.
- Hence determine the exact value of
$$\int _ { 0 } ^ { 2 } ( x - 2 ) ^ { 2 } e ^ { 4 x } d x$$