| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2016 |
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| Session | June |
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| Topic | Reduction Formulae |
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5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^ { n - 1 } x \sin ^ { 3 } x\) with respect to \(x\), prove that
$$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$
Hence find the exact value of \(I _ { 4 }\).