| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.8 This question requires partial fraction decomposition with an irreducible quadratic factor, followed by integration involving both logarithmic and arctangent forms. While the algebraic manipulation is systematic, the integration of the quadratic term requires recognizing standard forms and careful handling of the definite integral. This is moderately challenging for A-level, above average but not exceptional. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A = 3, B = 3, C = 0\) | A1 | |
| Obtain a second value | A1 | |
| Obtain a third value | A1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate and obtain term \(-3\ln(2 - x)\) | B1\(\sqrt{}\) | |
| Integrate and obtain term of the form \(k\ln(2 + x^2)\) | M1 | |
| Obtain term \(\frac{3}{2}\ln(2 + x^2)\) | A1\(\sqrt{}\) | |
| Substitute limits correctly in an integral of the form \(a\ln(2 - x) + b\ln(2 + x^2)\), where \(ab \neq 0\) | M1 | |
| Obtain given answer after full and correct working | A1 | 5 marks |
**(i)**
Use a correct method for finding a constant | M1 |
Obtain one of $A = 3, B = 3, C = 0$ | A1 |
Obtain a second value | A1 |
Obtain a third value | A1 | 4 marks
**(ii)**
Integrate and obtain term $-3\ln(2 - x)$ | B1$\sqrt{}$ |
Integrate and obtain term of the form $k\ln(2 + x^2)$ | M1 |
Obtain term $\frac{3}{2}\ln(2 + x^2)$ | A1$\sqrt{}$ |
Substitute limits correctly in an integral of the form $a\ln(2 - x) + b\ln(2 + x^2)$, where $ab \neq 0$ | M1 |
Obtain given answer after full and correct working | A1 | 5 marks8 Let $f ( x ) = \frac { 6 + 6 x } { ( 2 - x ) \left( 2 + x ^ { 2 } \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in the form $\frac { A } { 2 - x } + \frac { B x + C } { 2 + x ^ { 2 } }$.\\
(ii) Show that $\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 3 \ln 3$.
\hfill \mbox{\textit{CAIE P3 2014 Q8 [9]}}