| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Moderate -0.3 Part (a) is a straightforward reverse chain rule application with exponential functions requiring recognition that the derivative of e^(2x) appears as a factor. Part (b) requires the standard identity tan²θ = sec²θ - 1 to convert to integrable form, then evaluating a definite integral - this is a standard textbook exercise testing identity manipulation and basic integration. Both parts are routine applications of core techniques with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Rewrite integrand as \(12e^x + 4e^{3x}\) | B1 | |
| Integrate to obtain \(12e^x\) ... | B1 | |
| Integrate to obtain ... \(+ \frac{4}{3}e^{3x}\) | B1 | |
| Include ... \(+ c\) | B1 | [4] |
| (b) Use identity \(\tan^2\theta = \sec^2\theta - 1\) | B1 | |
| Integrate to obtain \(2\tan\theta + \theta\) or equivalent | B1 | |
| Use limits correctly for integral of form \(a\tan\theta + b\theta\) | M1 | |
| Confirm given answer \(\frac{1}{2}(8 + \pi)\) | A1 | [4] |
**(a)** Rewrite integrand as $12e^x + 4e^{3x}$ | B1 |
Integrate to obtain $12e^x$ ... | B1 |
Integrate to obtain ... $+ \frac{4}{3}e^{3x}$ | B1 |
Include ... $+ c$ | B1 | [4]
**(b)** Use identity $\tan^2\theta = \sec^2\theta - 1$ | B1 |
Integrate to obtain $2\tan\theta + \theta$ or equivalent | B1 |
Use limits correctly for integral of form $a\tan\theta + b\theta$ | M1 |
Confirm given answer $\frac{1}{2}(8 + \pi)$ | A1 | [4]6
\begin{enumerate}[label=(\alph*)]
\item Find $\int 4 \mathrm { e } ^ { x } \left( 3 + \mathrm { e } ^ { 2 x } \right) \mathrm { d } x$.
\item Show that $\int _ { - \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 4 } \pi } \left( 3 + 2 \tan ^ { 2 } \theta \right) \mathrm { d } \theta = \frac { 1 } { 2 } ( 8 + \pi )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2011 Q6 [8]}}