| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Difficulty | Standard +0.3 Part (a) requires recognizing the identity 1 + tan²(2x) = sec²(2x) to simplify before integrating, then applying reverse chain rule - a standard technique. Part (b) involves expanding sin(x + π/6) using compound angle formula, simplifying the quotient, then integrating - more steps but still routine application of known techniques. Both parts are slightly above average due to requiring identity manipulation before integration, but remain standard P3 exercises. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | ||
| Use identity \(\tan^2 2x = \sec^2 2x - 1\) | B1 | |
| Obtain integral of form \(ax + b\tan 2x\) | M1 | |
| Obtain correct \(3x + \frac{1}{2}\tan 2x\), condoning absence of \(+ c\) | A1 | [3] |
| (b) | ||
| State \(\sin x \cos \frac{1}{2}\pi + \cos x \sin -\frac{1}{6}\pi\) | B1 | |
| Simplify integrand to \(\cos\frac{1}{6}\pi + \frac{\cos x \sin \frac{1}{6}\pi}{\sin x}\) or equivalent | B1 | |
| Integrate to obtain at least term of form \(a\ln(\sin x)\) | *M1 | |
| Apply limits and simplify to obtain two terms | M1 dep *M | |
| Obtain \(\frac{1}{8}\pi\sqrt{3} - \frac{1}{2}\ln\left(\frac{1}{\sqrt{2}}\right)\) or equivalent | A1 | [5] |
**(a)** |
Use identity $\tan^2 2x = \sec^2 2x - 1$ | B1 |
Obtain integral of form $ax + b\tan 2x$ | M1 |
Obtain correct $3x + \frac{1}{2}\tan 2x$, condoning absence of $+ c$ | A1 | [3]
**(b)** |
State $\sin x \cos \frac{1}{2}\pi + \cos x \sin -\frac{1}{6}\pi$ | B1 |
Simplify integrand to $\cos\frac{1}{6}\pi + \frac{\cos x \sin \frac{1}{6}\pi}{\sin x}$ or equivalent | B1 |
Integrate to obtain at least term of form $a\ln(\sin x)$ | *M1 |
Apply limits and simplify to obtain two terms | M1 dep *M |
Obtain $\frac{1}{8}\pi\sqrt{3} - \frac{1}{2}\ln\left(\frac{1}{\sqrt{2}}\right)$ or equivalent | A1 | [5]5
\begin{enumerate}[label=(\alph*)]
\item Find $\int \left( 4 + \tan ^ { 2 } 2 x \right) \mathrm { d } x$.
\item Find the exact value of $\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \sin \left( x + \frac { 1 } { 6 } \pi \right) } { \sin x } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2015 Q5 [8]}}