| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with logarithmic form |
| Difficulty | Moderate -0.8 This is a straightforward application of the standard logarithmic integral formula with reverse chain rule. Part (i) requires recognizing the form ∫(f'(x)/f(x))dx = ln|f(x)| + c, and part (ii) is direct substitution into definite integral bounds with simple arithmetic to simplify ln(27) - ln(3) = ln(9). Both parts are routine exercises requiring only recall and basic manipulation, making this easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State indefinite integral of the form \(k \ln(4x - 1)\), where \(k = 2, 4,\) or \(\frac{1}{2}\) | M1 | |
| State correct integral \(\frac{1}{2} \ln(4x - 1)\) | A1 | [2] |
| (ii) Substitute limits correctly | M1 | |
| Use law for the logarithm of a power or a quotient | M1 | |
| Obtain \(\ln 3\) correctly | A1 | [3] |
(i) State indefinite integral of the form $k \ln(4x - 1)$, where $k = 2, 4,$ or $\frac{1}{2}$ | M1 |
State correct integral $\frac{1}{2} \ln(4x - 1)$ | A1 | [2]
(ii) Substitute limits correctly | M1 |
Use law for the logarithm of a power or a quotient | M1 |
Obtain $\ln 3$ correctly | A1 | [3]1 (i) Find $\int \frac { 2 } { 4 x - 1 } \mathrm {~d} x$.\\
(ii) Hence find $\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x$, expressing your answer in the form $\ln a$, where $a$ is an integer.
\hfill \mbox{\textit{CAIE P2 2013 Q1 [5]}}