| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with repeated linear factor |
| Difficulty | Standard +0.3 This is a standard partial fractions question with a repeated linear factor, followed by routine integration. The decomposition form is predictable (A/(2x+1) + B/(x+2) + C/(x+2)²), and the integration involves standard ln and power rules. While it requires careful algebra and multiple steps, it's a textbook exercise with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\dfrac{A}{2x+1} + \dfrac{B}{x+2} + \dfrac{C}{(x+2)^2}\) | B1 | Accept \(\dfrac{A}{2x+1} + \dfrac{Dx+E}{(x+2)^2}\) |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=1\), \(B=-2\), \(C=3\) | A1 | For alternative form: \(A=1\), \(D=-2\), \(E=-1\) |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate and obtain one of \(\frac{1}{2}\ln(2x+1)\), \(-2\ln(x+2)\), \(\dfrac{-3}{x+2}\) | B1 FT | The follow through is on *their* \(A\), \(B\), \(C\). |
| Obtain a second term | B1 FT | If the alternative form is used, then either need to use integration by parts or split the fraction further. |
| Obtain the third term | B1 FT | |
| Substitute limits correctly in an integral with at least two terms of the form \(\frac{1}{2}\ln(2x+1)\), \(-2\ln(x+2)\) and \(\dfrac{-3}{x+2}\) and subtract in correct order | M1 | The terms used need to have been obtained correctly. Must be exact values, not decimals. |
| Obtain \(1 - \ln 3\) | A1 | |
| Total | 5 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\dfrac{A}{2x+1} + \dfrac{B}{x+2} + \dfrac{C}{(x+2)^2}$ | B1 | Accept $\dfrac{A}{2x+1} + \dfrac{Dx+E}{(x+2)^2}$ |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=1$, $B=-2$, $C=3$ | A1 | For alternative form: $A=1$, $D=-2$, $E=-1$ |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | |
---
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain one of $\frac{1}{2}\ln(2x+1)$, $-2\ln(x+2)$, $\dfrac{-3}{x+2}$ | B1 FT | The follow through is on *their* $A$, $B$, $C$. |
| Obtain a second term | B1 FT | If the alternative form is used, then either need to use integration by parts or split the fraction further. |
| Obtain the third term | B1 FT | |
| Substitute limits correctly in an integral with at least two terms of the form $\frac{1}{2}\ln(2x+1)$, $-2\ln(x+2)$ and $\dfrac{-3}{x+2}$ and subtract in correct order | M1 | The terms used need to have been obtained correctly. Must be exact values, not decimals. |
| Obtain $1 - \ln 3$ | A1 | |
| **Total** | **5** | |
---8 Let $\mathrm { f } ( x ) = \frac { 3 - 3 x ^ { 2 } } { ( 2 x + 1 ) ( x + 2 ) ^ { 2 } }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find the exact value of $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x$, giving your answer in the form $a + b \ln c$, where $a , b$ and $c$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q8 [10]}}