6 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\).
- Prove that, for \(n \geqslant 2\),
$$I _ { n } + n ( n - 1 ) I _ { n - 2 } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } .$$
- Calculate the exact value of \(I _ { 1 }\) and deduce the exact value of \(I _ { 3 }\).