| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with exponentials |
| Difficulty | Moderate -0.8 Part (a) is a straightforward definite integral of sin with a linear argument requiring reverse chain rule (divide by coefficient). Part (b) requires expanding the brackets then integrating two standard exponential terms. Both are routine applications of basic integration techniques with no problem-solving or insight required, making this easier than average. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Obtain integral form of \(k\cos\frac{1}{2}x\) | M1 | |
| Obtain correct \(-2\cos\frac{1}{2}x\) | A1 | |
| Use limits correctly to obtain \(1\) | A1 | [3] |
| (b) Rewrite integrand as \(e^x + 1\) | B1 | |
| Integrate to obtain \(-e^{-x}\) ... | B1 | |
| Integrate to obtain ... \(+x + c\) | B1 | [3] |
**(a)** Obtain integral form of $k\cos\frac{1}{2}x$ | M1 |
Obtain correct $-2\cos\frac{1}{2}x$ | A1 |
Use limits correctly to obtain $1$ | A1 | [3]
**(b)** Rewrite integrand as $e^x + 1$ | B1 |
Integrate to obtain $-e^{-x}$ ... | B1 |
Integrate to obtain ... $+x + c$ | B1 | [3]4
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x$.
\item Find $\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2011 Q4 [6]}}