5. Given that
$$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$
- show that, for \(n \geqslant 1\)
$$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$
where \(p\) and \(q\) are constants to be found.
- Use part (a) to find the exact value of
$$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$
giving your answer in the form \(k \sqrt { 2 }\), where \(k\) is rational.