| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Derive reduction formula by integration by parts |
| Difficulty | Challenging +1.2 This is a standard reduction formula question requiring integration by parts with clear substitution (u = x^(n+1), dv = x·e^(-x²)dx). Part (i) is direct integration, part (ii) follows a well-established technique, and part (iii) applies the formula recursively. While it requires careful algebraic manipulation and is typical Further Maths content, it follows a predictable pattern without requiring novel insight. |
| Spec | 1.08i Integration by parts4.08g Derivatives: inverse trig and hyperbolic functions8.06a Reduction formulae: establish, use, and evaluate recursively |
4 It is given that, for $n \geqslant 0$,
$$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$
(i) Find $I _ { 1 }$ in terms of c .\\
(ii) Show that
$$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
(iii) Find $I _ { 5 }$ in terms of $e$.
\hfill \mbox{\textit{CAIE FP1 2002 Q4 [7]}}