3 The integral \(I _ { n }\), where \(n\) is a positive integer, is defined by
$$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$
- Show that
$$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$
- Find \(I _ { 5 }\) in terms of \(\pi\) and \(I _ { 1 }\).