| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Trigonometric substitution: show transformation then evaluate |
| Difficulty | Standard +0.3 This is a guided two-part question where part (i) walks students through a standard trigonometric substitution with clear instructions, and part (ii) requires the routine application of the double angle formula to integrate cos²θ. The substitution mechanics are straightforward (using 1+tan²θ = sec²θ), and the final integration is a standard textbook technique. Slightly easier than average due to the scaffolding provided. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(\frac{dx}{d\theta} = 2\sec^2\theta\) or \(dx = 2\sec^2\theta d\theta\) | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Obtain any correct form in terms of \(\theta\) | A1 | |
| Obtain the given form correctly (including the limits) | A1 | [4] |
| (ii) Use cos 2A formula, replacing integrand by \(a + b\cos 2\theta\), where \(ab \neq 0\) | M1* | |
| Integrate and obtain \(\frac{1}{4}\theta + \frac{1}{4}\sin 2\theta\) | A1 | |
| Use limits \(\theta = 0\) and \(\theta = \frac{1}{4}\pi\) | M1(dep*) | |
| Obtain answer \(\frac{1}{8}(\pi + 2)\), or exact equivalent | A1 | [4] |
**(i)** State or imply $\frac{dx}{d\theta} = 2\sec^2\theta$ or $dx = 2\sec^2\theta d\theta$ | B1 |
Substitute for $x$ and $dx$ throughout | M1 |
Obtain any correct form in terms of $\theta$ | A1 |
Obtain the given form correctly (including the limits) | A1 | [4]
**(ii)** Use cos 2A formula, replacing integrand by $a + b\cos 2\theta$, where $ab \neq 0$ | M1* |
Integrate and obtain $\frac{1}{4}\theta + \frac{1}{4}\sin 2\theta$ | A1 |
Use limits $\theta = 0$ and $\theta = \frac{1}{4}\pi$ | M1(dep*) |
Obtain answer $\frac{1}{8}(\pi + 2)$, or exact equivalent | A1 | [4]6 (i) Use the substitution $x = 2 \tan \theta$ to show that
$$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$$
(ii) Hence find the exact value of
$$\int _ { 0 } ^ { 2 } \frac { 8 } { \left( 4 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$$
\hfill \mbox{\textit{CAIE P3 2009 Q6 [8]}}