| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Trigonometric substitution to simplify integral |
| Difficulty | Standard +0.3 This is a structured question with clear guidance at each step. Part (i) is routine differentiation using quotient rule followed by chain rule - standard A-level techniques. Part (ii) provides the substitution explicitly and requires standard trigonometric manipulation to reach a logarithmic answer. While it involves multiple steps and careful algebra, the path is well-signposted and uses familiar techniques, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct quotient or chain rule to differentiate \(\sec x\) | M1 | |
| Obtain given derivative, \(\sec x \tan x\), correctly | A1 | |
| Use chain rule to differentiate \(y\) | M1 | |
| Obtain the given answer | A1 | [4] |
| (ii) Using \(dr\sqrt{3}\sec^2\theta \, d\theta\), or equivalently, express integral in terms of \(\theta\) and \(d\theta\) | M1 | |
| Obtain \(\int\sec\theta \, d\theta\) | A1 | |
| Use limits \(\frac{1}{6}\pi\) and \(\frac{1}{3}\pi\) correctly in an integral form of the type \(k\ln(\sec\theta + \tan\theta)\) | M1 | |
| Obtain a correct exact final answer in the given form, e.g. \(\ln\left(\frac{2 + \sqrt{3}}{\sqrt{3}}\right)\) | A1 | [4] |
**(i)** Use correct quotient or chain rule to differentiate $\sec x$ | M1 |
Obtain given derivative, $\sec x \tan x$, correctly | A1 |
Use chain rule to differentiate $y$ | M1 |
Obtain the given answer | A1 | [4]
**(ii)** Using $dr\sqrt{3}\sec^2\theta \, d\theta$, or equivalently, express integral in terms of $\theta$ and $d\theta$ | M1 |
Obtain $\int\sec\theta \, d\theta$ | A1 |
Use limits $\frac{1}{6}\pi$ and $\frac{1}{3}\pi$ correctly in an integral form of the type $k\ln(\sec\theta + \tan\theta)$ | M1 |
Obtain a correct exact final answer in the given form, e.g. $\ln\left(\frac{2 + \sqrt{3}}{\sqrt{3}}\right)$ | A1 | [4]\begin{enumerate}[label=(\roman*)]
\item By differentiating $\frac{1}{\cos x}$, show that the derivative of $\sec x$ is $\sec x \tan x$. Hence show that if $y = \ln(\sec x + \tan x)$ then $\frac{dy}{dx} = \sec x$. [4]
\item Using the substitution $x = (\sqrt{3}) \tan \theta$, find the exact value of
$$\int_1^3 \frac{1}{\sqrt{(3 + x^2)}} dx,$$
expressing your answer as a single logarithm. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2013 Q6 [8]}}