| Exam Board | Edexcel |
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| Module | C34 (Core Mathematics 3 & 4) |
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| Year | 2018 |
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| Session | October |
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| Topic | Integration by Parts |
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8. (i) Find \(\int x \sin x d x\)
(ii) (a) Use the substitution \(x = \sec \theta\) to show that
(b) Hence find the exact value of
$$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { \pi } { 3 } } \tan ^ { 2 } \theta \mathrm {~d} \theta$$
Hence find the exact value of
$$\int _ { 1 } ^ { 2 } \sqrt { 1 - \frac { 1 } { x ^ { 2 } } } \mathrm {~d} x$$