6 Let \(I _ { n }\) denote \(\int _ { 0 } ^ { 2 } \left( 4 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x\).
- Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - n } \right)\) and hence show that
$$8 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 \times 8 ^ { - n } .$$
- Use the result for integrating \(\frac { 1 } { x ^ { 2 } + a ^ { 2 } }\) with respect to \(x\), in the List of Formulae (MF10), to find the value of \(I _ { 1 }\) and deduce that
$$I _ { 3 } = \frac { 3 } { 1024 } \pi + \frac { 1 } { 128 }$$