5.
$$I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } ( 2 x - 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x , \quad n \geqslant 0$$
- Prove that, for \(n \geqslant 1\),
$$( 2 n + 1 ) I _ { n } = n I _ { n - 1 } + 3 \times 5 ^ { n } - 1$$
- Using the reduction formula given in part (a), find the exact value of \(I _ { 2 }\)