Standard +0.8 This is a multi-step integration by substitution requiring students to find du/dx, rearrange to express x² dx in terms of u, substitute into the integral, expand and integrate term-by-term, then back-substitute and factor to match the given form. While the substitution is provided, executing it correctly and algebraically manipulating to the required form with constants k and A requires solid technique and careful algebra—moderately above average difficulty.
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)
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)\\
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q13 [4]}}