| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with logarithmic form |
| Difficulty | Moderate -0.8 Part (a) is a straightforward application of exponential integration with a linear coefficient in the exponent. Part (b) is a standard logarithmic integration requiring recognition of the derivative of (3x-1) and simple arithmetic to verify the given answer. Both parts are routine exercises testing basic integration techniques with no problem-solving or novel insight required, 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 |
|---|---|
| (a) Obtain integral \(ke^{\frac{x}{2}}\) with any non-zero \(k\) | M1 |
| Correct integral | A1 |
| [2] | |
| (b) State indefinite integral of the form \(k \ln(3x - 1)\), where \(k = 2, 6\) or \(3\) | M1 |
| State correct integral \(2 \ln(3x - 1)\) | A1 |
| Substitute limits correctly (must be a function involving a logarithm) | M1 |
| Use law for the logarithm of a power or a quotient | M1 |
| Obtain given answer correctly | A1 |
| [5] |
**(a)** Obtain integral $ke^{\frac{x}{2}}$ with any non-zero $k$ | M1 |
Correct integral | A1 |
| | [2] |
**(b)** State indefinite integral of the form $k \ln(3x - 1)$, where $k = 2, 6$ or $3$ | M1 |
State correct integral $2 \ln(3x - 1)$ | A1 |
Substitute limits correctly (must be a function involving a logarithm) | M1 |
Use law for the logarithm of a power or a quotient | M1 |
Obtain given answer correctly | A1 |
| | [5] |6
\begin{enumerate}[label=(\alph*)]
\item Find $\int 4 \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x$.
\item Show that $\int _ { 1 } ^ { 3 } \frac { 6 } { 3 x - 1 } \mathrm {~d} x = \ln 16$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2012 Q6 [7]}}