| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Reduction formula or recurrence |
| Difficulty | Challenging +1.3 This is a standard FP3 reduction formula question with clear structure: integrate a simple expression, derive a recurrence relation using integration by parts, then apply it iteratively. While it requires multiple techniques (substitution, integration by parts, manipulation), the steps are well-signposted and follow a familiar pattern for Further Maths students. The algebra is straightforward and the final calculation is routine application of the formula. |
| Spec | 1.08h Integration by substitution8.06a Reduction formulae: establish, use, and evaluate recursively |
5.
$$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5$.
\item Using your answer to part (a), or otherwise, show that
$$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
\item Find $I _ { 4 }$ in the form $k \pi$, where $k$ is a fraction.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q5 [4]}}