| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | October |
| Marks | 8 |
| 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 straightforward partial fractions integration problem with one minor twist: the numerator degree equals the denominator degree, requiring polynomial division first. However, this is a standard technique taught in C4, and the subsequent partial fractions decomposition and integration are routine. The question is slightly easier than average because the algebra is clean, the limits are simple integers, and the final form is clearly specified. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2 + \ldots\) | B1 | |
| Obtains \(\frac{A}{x} + \frac{B}{x-1}\) where \(A\), \(B\) are constants | M1 | |
| \(\frac{3}{x}\) or \(-\frac{1}{x-1}\) or \(A=3\) or \(B=-1\) | A1 | One correct constant |
| \(\frac{3}{x} - \frac{1}{x-1}\) | A1 (B1 on Epen) | |
| \(\int_3^4 \frac{2x^2-3}{x(x-1)}\,dx = \int_3^4\left(2 + \frac{3}{x} - \frac{1}{x-1}\right)dx\) | ||
| \(= \left[2x + 3\ln x - \ln(x-1)\right]_3^4\) | M1 A1ft | |
| \(= (8+3\ln 4 - \ln 3)-(6+3\ln 3-\ln 2) = 2+\ln\!\left(\frac{128}{81}\right)\) | M1 A1cso | (8 marks) |
# Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2 + \ldots$ | B1 | |
| Obtains $\frac{A}{x} + \frac{B}{x-1}$ where $A$, $B$ are constants | M1 | |
| $\frac{3}{x}$ or $-\frac{1}{x-1}$ or $A=3$ or $B=-1$ | A1 | One correct constant |
| $\frac{3}{x} - \frac{1}{x-1}$ | A1 **(B1 on Epen)** | |
| $\int_3^4 \frac{2x^2-3}{x(x-1)}\,dx = \int_3^4\left(2 + \frac{3}{x} - \frac{1}{x-1}\right)dx$ | | |
| $= \left[2x + 3\ln x - \ln(x-1)\right]_3^4$ | M1 A1ft | |
| $= (8+3\ln 4 - \ln 3)-(6+3\ln 3-\ln 2) = 2+\ln\!\left(\frac{128}{81}\right)$ | M1 A1cso | **(8 marks)** |
**Notes:**
- B1: $2+\ldots$
- M1: Obtains $\frac{A}{x}+\frac{B}{x-1}$
- A1: $\frac{3}{x}$ or $-\frac{1}{x-1}$ or one correct constant
- B1: $\frac{3}{x}-\frac{1}{x-1}$
- M1: For $\int \frac{*}{x}+\frac{*}{x-1}\,dx \to p\ln mx + q\ln n(x-1)$
- A1ft: $2x+3\ln x - \ln(x-1)$; follow through on "2", $A$ and $B$
- M1: Substituting 3 and 4, subtracting, using correct log laws at least once
- A1cso: $2+\ln\!\left(\frac{128}{81}\right)$ or $2+\ln\!\left(1\frac{47}{81}\right)$; do not allow $2+\ln\!\left(\frac{2^7}{3^4}\right)$; $2+\ln\!\left(\frac{128}{81}\right)+c$ is A0
---8. Use partial fractions, and integration, to find the exact value of $\int _ { 3 } ^ { 4 } \frac { 2 x ^ { 2 } - 3 } { x ( x - 1 ) } \mathrm { d } x$
Write your answer in the form $a + \ln b$, where $a$ is an integer and $b$ is a rational constant.
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\hfill \mbox{\textit{Edexcel C34 2017 Q8 [8]}}