| Exam Board | CAIE |
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| Module | FP1 (Further Pure Mathematics 1) |
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| Year | 2011 |
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| Session | June |
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| Topic | Reduction Formulae |
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5 Let
$$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$
where \(n \geqslant 0\). Use the fact that \(\tan ^ { 2 } x = \sec ^ { 2 } x - 1\) to show that, for \(n \geqslant 2\),
$$I _ { n } = \frac { 1 } { n - 1 } - I _ { n - 2 }$$
Show that \(I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi\).