| Exam Board | Edexcel |
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| Module | C4 (Core Mathematics 4) |
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| Topic | Integration by Parts |
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14. (i) Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
(ii) Use the substitution \(x = \sin \theta\) to show that, for \(| x | \leq 1\), \(\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c\), where \(c\) is an arbitrary constant.