| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with logarithmic form |
| Difficulty | Moderate -0.3 This is a straightforward application of standard integrals. Part (a) requires integrating sin(2x) using reverse chain rule and sec²(x) by recall, then evaluating at limits. Part (b) integrates 1/x forms to logarithms and simplifies using log laws. Both parts are routine exercises testing basic integration techniques with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks |
|---|---|
| Integrate and obtain term \(k \cos 2x\), where \(k = \pm\frac{1}{2}\) or \(+1\) | M1 |
| Obtain term \(-\frac{1}{2}\cos 2x\) | A1 |
| Obtain term \(\tan x\) | B1 |
| Substitute correct limits correctly | M1 |
| Obtain exact answer \(\frac{3}{4} + \sqrt{3}\) | A1 |
| Answer | Marks |
|---|---|
| Integrate and obtain \(\frac{1}{2}\ln x + \ln(x + 1)\) or \(\frac{1}{2}\ln 2x + \ln(x + 1)\) | B1 + B1 |
| Substitute correct limits correctly | M1 |
| Obtain given answer following full and correct working | A1 |
**(a)**
| Integrate and obtain term $k \cos 2x$, where $k = \pm\frac{1}{2}$ or $+1$ | M1 |
| Obtain term $-\frac{1}{2}\cos 2x$ | A1 |
| Obtain term $\tan x$ | B1 |
| Substitute correct limits correctly | M1 |
| Obtain exact answer $\frac{3}{4} + \sqrt{3}$ | A1 |
**Total: [5]**
**(b)**
| Integrate and obtain $\frac{1}{2}\ln x + \ln(x + 1)$ or $\frac{1}{2}\ln 2x + \ln(x + 1)$ | B1 + B1 |
| Substitute correct limits correctly | M1 |
| Obtain given answer following full and correct working | A1 |
**Total: [4]**8
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \sin 2 x + \sec ^ { 2 } x \right) \mathrm { d } x$.
\item Show that $\int _ { 1 } ^ { 4 } \left( \frac { 1 } { 2 x } + \frac { 1 } { x + 1 } \right) \mathrm { d } x = \ln 5$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2009 Q8 [9]}}