7.
$$I _ { n } = \int \cosh ^ { n } 2 x \mathrm {~d} x \quad n \geqslant 0$$
- Show that, for \(n \geqslant 2\)
$$I _ { n } = \frac { \cosh ^ { n - 1 } 2 x \sinh 2 x } { 2 n } + \frac { n - 1 } { n } I _ { n - 2 }$$
- Hence determine
$$\int ( 1 + \cosh 2 x ) ^ { 3 } d x$$
collecting any like terms in your answer.