| Exam Board | SPS |
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| Module | SPS SM Pure (SPS SM Pure) |
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| Year | 2025 |
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| Session | February |
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| Topic | Integration by Substitution |
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13. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that
$$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + C$$
where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
(Total for Question 13 is 4 marks)