| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Standard +0.3 Part (a) requires recognizing to split the fraction and use sec²(2x) identity, then applying reverse chain rule - a standard technique. Part (b) is routine integration of polynomial and 1/x form with substitution, followed by logarithm manipulation. Both parts test standard A-level integration techniques with minimal problem-solving required, making this slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Rewrite integrand as \(\sec^2 2x + \cos^2 2x\) | B1 | |
| Express \(\cos^2 2x\) in form \(k_1 + k_2\cos 4x\) | M1 | |
| State correct \(\frac{1}{2} + \frac{1}{2}\cos 4x\) | A1 | |
| Integrate to obtain at least terms involving \(\tan 2x\) and \(\sin 4x\) | M1 | |
| Obtain \(\frac{1}{2}\tan 2x + \frac{3}{8}x + \frac{1}{8}\sin 4x\), condoning absence of \(+c\) | A1 | [5] |
| (b) Integrate to obtain \(2x + 2\ln(3x-2)\) | B1 | |
| Show correct use of \(p\ln k = \ln k^p\) law at least once, must be using \(\ln(3x-2)\) | M1 | |
| Show correct use of \(\ln m - \ln n = \ln\frac{m}{n}\) law, must be using \(\ln(3x-2)\) | M1 | |
| Use or imply \(20 = \ln(e^{20})\) | B1 | |
| Obtain \(\ln(16e^{20})\) | A1 | [5] |
(a) Rewrite integrand as $\sec^2 2x + \cos^2 2x$ | B1 |
Express $\cos^2 2x$ in form $k_1 + k_2\cos 4x$ | M1 |
State correct $\frac{1}{2} + \frac{1}{2}\cos 4x$ | A1 |
Integrate to obtain at least terms involving $\tan 2x$ and $\sin 4x$ | M1 |
Obtain $\frac{1}{2}\tan 2x + \frac{3}{8}x + \frac{1}{8}\sin 4x$, condoning absence of $+c$ | A1 | [5]
(b) Integrate to obtain $2x + 2\ln(3x-2)$ | B1 |
Show correct use of $p\ln k = \ln k^p$ law at least once, must be using $\ln(3x-2)$ | M1 |
Show correct use of $\ln m - \ln n = \ln\frac{m}{n}$ law, must be using $\ln(3x-2)$ | M1 |
Use or imply $20 = \ln(e^{20})$ | B1 |
Obtain $\ln(16e^{20})$ | A1 | [5]7
\begin{enumerate}[label=(\alph*)]
\item Find $\int \frac { 1 + \cos ^ { 4 } 2 x } { \cos ^ { 2 } 2 x } \mathrm {~d} x$.
\item Without using a calculator, find the exact value of $\int _ { 4 } ^ { 14 } \left( 2 + \frac { 6 } { 3 x - 2 } \right) \mathrm { d } x$, giving your answer in the form $\ln \left( a \mathrm { e } ^ { b } \right)$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2016 Q7 [10]}}