6.
$$I _ { n } = \int \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0$$
- Show that
$$I _ { n } = \frac { \mathrm { e } ^ { x } \sin ^ { n - 1 } x } { n ^ { 2 } + 1 } ( \sin x - n \cos x ) + \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 } \quad n \geqslant 2$$
- Hence find the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } e ^ { x } \sin ^ { 4 } x d x$$
giving your answer in the form \(A \mathrm { e } ^ { \frac { \pi } { 2 } } + B\) where \(A\) and \(B\) are rational numbers to be determined.