Challenging +1.2 This question requires integration by parts to derive the reduction formula, followed by a straightforward induction proof. While it involves multiple techniques (integration by parts, reduction formulae, and induction), each step follows standard procedures without requiring novel insight. The induction is mechanical once the reduction formula is established, making this moderately above average but not exceptionally challenging for Further Maths students.
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 [9]}}