| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2013 |
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| Session | June |
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| Topic | Reduction Formulae |
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4 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x\). Prove that, for every positive integer \(n\),
$$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$
Given that \(I _ { 1 } = \frac { 1 } { 4 } \pi\), find the exact value of \(I _ { 3 }\).