6.
$$I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x \quad n \in \mathbb { N }$$
- Show that
$$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } - \frac { 3 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 3$$
- Hence show that
$$\int \frac { x ^ { 5 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 1 } { 5 } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } \left( x ^ { 4 } + p x ^ { 2 } + q \right) + k$$
where \(p\) and \(q\) are integers to be determined and \(k\) is an arbitrary constant.