Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that
$$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$
where \(c\) is an arbitrary constant.
$$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
Show that, for \(n \geqslant 2\)
$$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that