Challenging +1.8 This is a multi-part reduction formula question requiring integration by parts to establish the recurrence relation, recursive calculation to find I₄, and then geometric reasoning to establish bounds on e. While the integration by parts is standard, the final part requires insight to connect the integral to bounds on e by comparing with the area under y=(1-x)⁴, which is non-routine problem-solving for A-level students.
7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\),
$$I _ { n } = n I _ { n - 1 } - 1$$
Find the exact value of \(I _ { 4 }\).
By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that
$$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$
7 Let $I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x$. Show that, for all positive integers $n$,
$$I _ { n } = n I _ { n - 1 } - 1$$
Find the exact value of $I _ { 4 }$.
By considering the area of the region enclosed by the $x$-axis, the $y$-axis and the curve with equation $y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }$ in the interval $0 \leqslant x \leqslant 1$, show that
$$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$
\hfill \mbox{\textit{CAIE FP1 2014 Q7 [10]}}