15.
$$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x \text { and } J _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x , \quad n \geq 0$$
- Show that, for \(n \geq 1\),
$$I _ { n } = \mathrm { e } - n I _ { n - 1 } .$$
- Find a similar reduction formula for \(J _ { n }\).
- Show that \(J _ { 2 } = 2 - \frac { 5 } { \mathrm { e } }\).
- Show that \(\int _ { 0 } ^ { 1 } x ^ { n } \cosh x \mathrm {~d} x = \frac { 1 } { 2 } \left( I _ { n } + J _ { n } \right)\).
- Hence, or otherwise, evaluate \(\int _ { 0 } ^ { 1 } x ^ { 2 } \cosh x \mathrm {~d} x\), giving your answer in terms of e.
[0pt]
[P5 June 2003 Qn 7]