| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2013 |
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| Session | November |
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| Topic | Reduction Formulae |
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Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { n } \mathrm {~d} x\). Show that, for all integers \(n\),
$$( 2 n + 1 ) I _ { n } = 2 n I _ { n - 1 } + 2 ^ { n }$$
Evaluate \(I _ { 0 }\) and hence find \(I _ { 3 }\).
Given that \(I _ { - 1 } = \frac { 1 } { 4 } \pi\), find \(I _ { - 3 }\).