| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Standard +0.3 This is a standard two-part question combining algebraic division with partial fractions, followed by routine integration. Part (a) requires polynomial long division then partial fraction decomposition—a core C4 technique but mechanical once learned. Part (b) applies the result to integrate term-by-term using standard logarithm rules. While it has 10 marks total and multiple steps, each step follows a well-practiced algorithm with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x^3 - 5x^2 + 6 = (Ax + B)(x - 3) + C(x - 3) + Dx\) | M1 | |
| \(x = 0 \Rightarrow 6 = -3C\) | A1 | |
| \(x = 3 \Rightarrow 15 = 3D\) | B1 | |
| coeffs \(x^3 \Rightarrow A = 2\) | M1 A1 | |
| coeffs \(x^2 \Rightarrow -5 = B - 3A\) | M1 A1 | |
| \(C = -2\), \(B = 1\) | A1 | |
| \(= \int_1^2 (2x + 1 - \frac{2}{x} + \frac{5}{x-3})dx\) | M1 | |
| \(= [x^2 + x - 2\ln | x | + 5\ln |
| \(= (4 + 2 - 2\ln 2 + 0) - (1 + 1 + 0 + 5\ln 2)\) | M1 | |
| \(= 4 - 7\ln 2\) | A1 | (10 marks) |
$2x^3 - 5x^2 + 6 = (Ax + B)(x - 3) + C(x - 3) + Dx$ | M1 |
$x = 0 \Rightarrow 6 = -3C$ | A1 |
$x = 3 \Rightarrow 15 = 3D$ | B1 |
coeffs $x^3 \Rightarrow A = 2$ | M1 A1 |
coeffs $x^2 \Rightarrow -5 = B - 3A$ | M1 A1 |
$C = -2$, $B = 1$ | A1 |
$= \int_1^2 (2x + 1 - \frac{2}{x} + \frac{5}{x-3})dx$ | M1 |
$= [x^2 + x - 2\ln|x| + 5\ln|x-3|]_1^2$ | M1 A2 |
$= (4 + 2 - 2\ln 2 + 0) - (1 + 1 + 0 + 5\ln 2)$ | M1 |
$= 4 - 7\ln 2$ | A1 | (10 marks)
---\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $A$, $B$, $C$ and $D$ such that
$$\frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \equiv Ax + B + \frac{C}{x} + \frac{D}{x-3}.$$ [5]
\item Evaluate
$$\int_1^2 \frac{2x^3 - 5x^2 + 6}{x^2 - 3x} \, dx,$$
giving your answer in the form $p + q \ln 2$, where $p$ and $q$ are integers. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [10]}}