| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with logarithmic form |
| Difficulty | Moderate -0.3 Part (a) is a straightforward definite integral requiring recognition of standard logarithmic forms (∫1/x dx = ln|x| and ∫1/(ax+b) dx = (1/a)ln|ax+b|), then substituting limits and simplifying to the given answer. Part (b) requires rewriting sin 2x as 2sin x cos x, then recognizing the resulting expression simplifies to a standard form, but this is still routine manipulation. Both parts are standard textbook exercises with no novel insight required, making this 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 | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k_1 \ln x + k_2 \ln(2x+1)\) | M1 | |
| Obtain correct \(2\ln x + \ln(2x+1)\) | A1 | |
| Use logarithm addition/subtraction property correctly | M1 | |
| Use logarithm power property correctly | M1 | |
| Confirm \(\ln 48\) with no errors seen | A1 | Answer given; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use identity \(\sin 2x = 2\sin x \cos x\) | B1 | |
| State or imply \(\cot x + 2\cosec x = \dfrac{\cos x}{\sin x} + \dfrac{2}{\sin x}\) | B1 | |
| Attempt to express integrand in terms of \(\cos 2x\) and \(\cos x\) | M1 | |
| Obtain correct integrand \(1 + \cos 2x + 4\cos x\) | A1 | |
| Integrate to obtain at least terms \(k_3 \sin 2x\) and \(k_4 \sin x\) | M1 | |
| Obtain correct \(x + \frac{1}{2}\sin 2x + 4\sin x + c\) | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k_1 \ln x + k_2 \ln(2x+1)$ | M1 | |
| Obtain correct $2\ln x + \ln(2x+1)$ | A1 | |
| Use logarithm addition/subtraction property correctly | M1 | |
| Use logarithm power property correctly | M1 | |
| Confirm $\ln 48$ with no errors seen | A1 | Answer given; necessary detail needed |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use identity $\sin 2x = 2\sin x \cos x$ | B1 | |
| State or imply $\cot x + 2\cosec x = \dfrac{\cos x}{\sin x} + \dfrac{2}{\sin x}$ | B1 | |
| Attempt to express integrand in terms of $\cos 2x$ and $\cos x$ | M1 | |
| Obtain correct integrand $1 + \cos 2x + 4\cos x$ | A1 | |
| Integrate to obtain at least terms $k_3 \sin 2x$ and $k_4 \sin x$ | M1 | |
| Obtain correct $x + \frac{1}{2}\sin 2x + 4\sin x + c$ | A1 | |
---6
\begin{enumerate}[label=(\alph*)]
\item Show that $\int _ { 1 } ^ { 4 } \left( \frac { 2 } { x } + \frac { 2 } { 2 x + 1 } \right) \mathrm { d } x = \ln 48$.
\item Find $\int \sin 2 x ( \cot x + 2 \operatorname { cosec } x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2019 Q6 [11]}}