5 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { n } \mathrm {~d} x$$ where \(n \geqslant 1\). Show that $$I _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } ( n + 1 ) I _ { n }$$ Hence prove by induction that, for all positive integers \(n , I _ { n }\) is of the form \(A _ { n } \mathrm { e } ^ { 2 } + B _ { n }\), where \(A _ { n }\) and \(B _ { n }\) are rational numbers.

5 Let

$$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { n } \mathrm {~d} x$$

where $n \geqslant 1$. Show that

$$I _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } ( n + 1 ) I _ { n }$$

Hence prove by induction that, for all positive integers $n , I _ { n }$ is of the form $A _ { n } \mathrm { e } ^ { 2 } + B _ { n }$, where $A _ { n }$ and $B _ { n }$ are rational numbers.

\hfill \mbox{\textit{CAIE FP1 2010 Q5}}