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}}