| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Rational function powers |
| Difficulty | Challenging +1.8 This is a substantial Further Maths reduction formula question requiring: (i) product rule differentiation and algebraic manipulation to establish a recurrence relation involving definite integrals, (ii) geometric reasoning about curve area, and (iii) iterative application of the formula. While the techniques are standard for FP1, the multi-step nature, need to connect differentiation with integration via limits, and final numerical verification make this significantly harder than typical A-level questions but not exceptionally difficult for Further Maths students who have practiced reduction formulae. |
| Spec | 1.07q Product and quotient rules: differentiation1.08d Evaluate definite integrals: between limits8.06a Reduction formulae: establish, use, and evaluate recursively |
9 It is given that
$$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$
where $n > 0$.\\
(i) Show that
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$
and hence, or otherwise, show that
$$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
(ii) By considering the graph of $y = \frac { 1 } { 1 + x ^ { 3 } }$, show that $I _ { 1 } < 1$.\\
(iii) Deduce that $I _ { 3 } < \frac { 53 } { 72 }$.
\hfill \mbox{\textit{CAIE FP1 2004 Q9 [10]}}