7 It is given that, for integers \(n \geqslant 1\),
$$I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$$
- Use integration by parts to show that \(I _ { n } = 2 ^ { - n } + 2 n \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) ^ { n + 1 } } \mathrm {~d} x\).
- Show that \(2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }\).
- Find \(I _ { 2 }\) in terms of \(\pi\).