| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2012 |
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| Session | June |
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| Topic | Reduction Formulae |
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4 Let
$$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x$$
for \(n \geqslant 0\). Show that, for all \(n \geqslant 1\),
$$I _ { n } = \frac { 1 } { 3 } \mathrm { e } ^ { 3 } - \frac { 1 } { 3 } n I _ { n - 1 }$$
Find the exact value of \(I _ { 3 }\).