| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Reduction formula or recurrence |
| Difficulty | Challenging +1.8 This is a Further Maths FP3 reduction formula question requiring integration by parts with algebraic manipulation to derive the recurrence relation, then apply it to find a specific integral. While the technique is standard for FP3, the algebraic manipulation needed to arrive at the exact form given is non-trivial and requires careful handling of terms. Part (b) requires iterating the formula and integrating I₁ = arctan(x), making this substantially harder than average A-level questions but routine for Further Maths students who have practiced reduction formulas. |
| Spec | 1.08i Integration by parts8.06a Reduction formulae: establish, use, and evaluate recursively |
9.
$$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that, for $n > 0$,
$$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
\item Find $I _ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q9 [8]}}