| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Reduction formula or recurrence |
| Difficulty | Challenging +1.2 This is a standard Further Maths reduction formula question requiring two applications of integration by parts to derive the recurrence relation, then systematic application to find a specific integral. While it requires careful algebraic manipulation and multiple steps, the technique is routine for F3 students and follows a well-practiced pattern with no novel insight needed. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08i Integration by parts8.06a Reduction formulae: establish, use, and evaluate recursively |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(\int x^n\cos x\,dx = x^n\sin x - \int nx^{n-1}\sin x\,dx\) | M1A1 | M1: Parts in the correct direction; A1: Correct expression |
| \(= x^n\sin x - \left\{-nx^{n-1}\cos x + \int n(n-1)x^{n-2}\cos x\,dx\right\}\) | dM1 | Uses integration by parts again (dependent on first M) |
| \(= x^n\sin x + nx^{n-1}\cos x - n(n-1)I_{n-2}\) | A1* | Fully correct proof with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(I_n = \int x^{n-1}(x\cos x)\,dx = x^n\sin x + x^{n-1}\cos x - (n-1)\int x^{n-2}(x\sin x+\cos x)\,dx\) | M1A1 | M1: Parts in correct direction; A1: Correct expression |
| \(= x^n\sin x + x^{n-1}\cos x - (n-1)\left\{-x^{n-1}\cos x + (n-1)I_{n-2}\right\} - (n-1)I_{n-2}\) | dM1 | Uses integration by parts again (dependent on first M) |
| \(= x^n\sin x + nx^{n-1}\cos x - n(n-1)I_{n-2}\) | A1* | Fully correct proof with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(I_0 = \sin x\ (+k)\) | B1 | |
| \(I_4 = x^4\sin x + 4x^3\cos x - 12I_2\) | M1 | Applies reduction formula once for \(I_4\) or \(I_2\) |
| \(= x^4\sin x + 4x^3\cos x - 12\left(x^2\sin x + 2x\cos x - 2I_0\right)\) | M1 | Applies reduction formula again and obtains expression for \(I_4\) which can include \(I_0\) but not \(I_2\) |
| \(= (x^4-12x^2+24)\sin x + (4x^3-24x)\cos x + c\) | A1A1 | Award A1 for either bracket and A1 for the other. If answer is not factorised but otherwise correct, award A1A0 |
# Question 4:
$$I_n = \int x^n \cos x\, dx$$
## Part (a):
| Working/Answer | Mark | Notes |
|---|---|---|
| $\int x^n\cos x\,dx = x^n\sin x - \int nx^{n-1}\sin x\,dx$ | M1A1 | M1: Parts in the correct direction; A1: Correct expression |
| $= x^n\sin x - \left\{-nx^{n-1}\cos x + \int n(n-1)x^{n-2}\cos x\,dx\right\}$ | dM1 | Uses integration by parts again (dependent on first M) |
| $= x^n\sin x + nx^{n-1}\cos x - n(n-1)I_{n-2}$ | A1* | Fully correct proof with no errors |
### Alternative:
| Working/Answer | Mark | Notes |
|---|---|---|
| $I_n = \int x^{n-1}(x\cos x)\,dx = x^n\sin x + x^{n-1}\cos x - (n-1)\int x^{n-2}(x\sin x+\cos x)\,dx$ | M1A1 | M1: Parts in correct direction; A1: Correct expression |
| $= x^n\sin x + x^{n-1}\cos x - (n-1)\left\{-x^{n-1}\cos x + (n-1)I_{n-2}\right\} - (n-1)I_{n-2}$ | dM1 | Uses integration by parts again (dependent on first M) |
| $= x^n\sin x + nx^{n-1}\cos x - n(n-1)I_{n-2}$ | A1* | Fully correct proof with no errors |
## Part (b):
| Working/Answer | Mark | Notes |
|---|---|---|
| $I_0 = \sin x\ (+k)$ | B1 | |
| $I_4 = x^4\sin x + 4x^3\cos x - 12I_2$ | M1 | Applies reduction formula once for $I_4$ or $I_2$ |
| $= x^4\sin x + 4x^3\cos x - 12\left(x^2\sin x + 2x\cos x - 2I_0\right)$ | M1 | Applies reduction formula again and obtains expression for $I_4$ which can include $I_0$ but not $I_2$ |
| $= (x^4-12x^2+24)\sin x + (4x^3-24x)\cos x + c$ | A1A1 | Award A1 for either bracket and A1 for the other. If answer is not factorised but otherwise correct, award A1A0 |
---4.
\begin{enumerate}[label=(\alph*)]
\item Show that, for $n \geqslant 2$
\item Hence find the functions $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ such that
$$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$
where $c$ is an arbitrary constant.
$$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
(a) Show that, for $n \geqslant 2$
$$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
(b) Hence find the functions $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ such that
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2020 Q4 [9]}}