| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Basic partial fractions then integrate |
| Difficulty | Moderate -0.3 This is a straightforward partial fractions question with a difference of squares denominator (16 - 9x²) that factors easily, followed by standard logarithmic integration. While it requires multiple steps (factorizing, finding partial fractions, integrating, combining logs), each step follows a well-practiced routine with no novel insight needed. Slightly easier than average due to the clean factorization and standard technique application. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\dfrac{9(4+x)}{16-9x^2} \equiv \dfrac{A}{(4-3x)} + \dfrac{B}{(4+3x)}\) and finds at least one of \(A\) or \(B\) | M1 | — |
| \(A = 6\) or \(B = 3\) obtained at any point | A1 | — |
| \(\dfrac{9(4+x)}{16-9x^2} \equiv \dfrac{6}{(4-3x)} + \dfrac{3}{(4+3x)}\) | A1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int \dfrac{9(4+x)}{16-9x^2}\,dx = -\dfrac{A}{3}\ln(4-3x) + \dfrac{B}{3}\ln(4+3x)\) | M1 A1ft | Integrates partial fractions to obtain \(\ldots\ln(4-3x) + \ldots\ln(4+3x)\); A1ft correct for their \(A,B\); modulus signs not required at this stage |
| Combines log terms with constant of integration: \(\ln\dfrac{(4+3x)}{(4-3x)^2} + c\) | M1 | For combining log terms correctly with \(c\) on same line |
| \(= \ln\dfrac{k(4+3x)}{(4-3x)^2}\) or \(\ln\left | \dfrac{k(4+3x)}{(4-3x)^2}\right | \) |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\dfrac{9(4+x)}{16-9x^2} \equiv \dfrac{A}{(4-3x)} + \dfrac{B}{(4+3x)}$ and finds at least one of $A$ or $B$ | M1 | — |
| $A = 6$ or $B = 3$ obtained at any point | A1 | — |
| $\dfrac{9(4+x)}{16-9x^2} \equiv \dfrac{6}{(4-3x)} + \dfrac{3}{(4+3x)}$ | A1 | — |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int \dfrac{9(4+x)}{16-9x^2}\,dx = -\dfrac{A}{3}\ln(4-3x) + \dfrac{B}{3}\ln(4+3x)$ | M1 A1ft | Integrates partial fractions to obtain $\ldots\ln(4-3x) + \ldots\ln(4+3x)$; A1ft correct for their $A,B$; modulus signs not required at this stage |
| Combines log terms with constant of integration: $\ln\dfrac{(4+3x)}{(4-3x)^2} + c$ | M1 | For combining log terms correctly with $c$ on same line |
| $= \ln\dfrac{k(4+3x)}{(4-3x)^2}$ or $\ln\left|\dfrac{k(4+3x)}{(4-3x)^2}\right|$ | A1 | cao |
---\begin{enumerate}
\item (a) Express $\frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } }$ in partial fractions.
\end{enumerate}
Given that
$$\mathrm { f } ( x ) = \frac { 9 ( 4 + x ) } { 16 - 9 x ^ { 2 } } , \quad x \in \mathbb { R } , \quad - \frac { 4 } { 3 } < x < \frac { 4 } { 3 }$$
(b) express $\int \mathrm { f } ( x ) \mathrm { d } x$ in the form $\ln ( \mathrm { g } ( x ) )$, where $\mathrm { g } ( x )$ is a rational function.\\
\hfill \mbox{\textit{Edexcel C34 2018 Q5 [7]}}