| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| 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 +0.8 This is a structured trigonometric substitution problem requiring careful manipulation of limits, differentials, and trigonometric identities. Part (a) requires executing the substitution x=3tan(θ) with proper handling of dx and simplification using sec²(θ) identities. Part (b) requires integrating cos²(θ) using the double angle formula and evaluating exactly. While methodical, it demands multiple techniques and careful algebraic manipulation beyond routine integration, placing it moderately above average difficulty. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(dx = 3\sec^2\theta \, d\theta\) | B1 | |
| Substitute throughout for \(x\) and \(dx\) | M1 | |
| Obtain any correct form in terms of \(\theta\) | A1 | e.g. \(\int \frac{81\sec^2\theta}{(9+9\tan^2\theta)^2} d\theta\) |
| Justify change of limits and obtain \(\int_0^{\frac{\pi}{4}}\cos^2\theta \, d\theta\) correctly | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain indefinite integral of the form \(\int a + b\cos 2\theta \, d\theta\), where \(ab \neq 0\) | *M1 | |
| Obtain \(\frac{1}{2}\theta + \frac{1}{4}\sin 2\theta\) | A1 | |
| Use correct limits correctly in an expression containing \(p\theta\) and \(q\sin 2\theta\) where \(pq \neq 0\) | DM1 | \(\frac{\pi}{8} + \frac{1}{4}(-0)\) |
| Obtain answer \(\frac{1}{8}(\pi + 2)\) | A1 | Or exact equivalent e.g. \(\frac{1}{8}\pi + \frac{1}{4}\) |
## Question 6:
### Part 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $dx = 3\sec^2\theta \, d\theta$ | B1 | |
| Substitute throughout for $x$ and $dx$ | M1 | |
| Obtain any correct form in terms of $\theta$ | A1 | e.g. $\int \frac{81\sec^2\theta}{(9+9\tan^2\theta)^2} d\theta$ |
| Justify change of limits and obtain $\int_0^{\frac{\pi}{4}}\cos^2\theta \, d\theta$ correctly | A1 | AG |
### Part 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain indefinite integral of the form $\int a + b\cos 2\theta \, d\theta$, where $ab \neq 0$ | *M1 | |
| Obtain $\frac{1}{2}\theta + \frac{1}{4}\sin 2\theta$ | A1 | |
| Use correct limits correctly in an expression containing $p\theta$ and $q\sin 2\theta$ where $pq \neq 0$ | DM1 | $\frac{\pi}{8} + \frac{1}{4}(-0)$ |
| Obtain answer $\frac{1}{8}(\pi + 2)$ | A1 | Or exact equivalent e.g. $\frac{1}{8}\pi + \frac{1}{4}$ |
---6 Let $I = \int _ { 0 } ^ { 3 } \frac { 27 } { \left( 9 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$.
\begin{enumerate}[label=(\alph*)]
\item Using the substitution $x = 3 \tan \theta$, show that $I = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$.
\item Hence find the exact value of $I$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q6 [8]}}