| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| 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 standard trigonometric substitution problem requiring multiple steps: applying the substitution correctly (including limits and dx), simplifying the radical, using the double angle formula for sin²θ, and integrating to find an exact answer. While methodical, it requires careful algebraic manipulation and knowledge of standard techniques, 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 |
|---|---|
| (i) State or imply \(\frac{dx}{d\theta} = 2\cos\theta \, d\theta\) or \(\frac{d\theta}{dx} = 2\cos\theta\), or equivalent | B1 |
| Substitute for \(x\) and \(dx\) throughout the integral | M1 |
| Obtain the given answer correctly, having changed limits and shown sufficient working | A1 |
| Answer | Marks |
|---|---|
| (ii) Replace integrand by \(2 - 2\cos 2\theta\), or equivalent | B1 |
| Obtain integral \(2\theta - \sin 2\theta\), or equivalent | B1√ |
| Substitute limits correctly in an integral of the form \(a\theta \pm b\sin 2\theta\), where \(ab \neq 0\) | M1 |
| Obtain answer \(\frac{1}{3}\pi - \frac{\sqrt{3}}{2}\), or exact equivalent | A1 |
**(i)** State or imply $\frac{dx}{d\theta} = 2\cos\theta \, d\theta$ or $\frac{d\theta}{dx} = 2\cos\theta$, or equivalent | B1 |
Substitute for $x$ and $dx$ throughout the integral | M1 |
Obtain the given answer correctly, having changed limits and shown sufficient working | A1 |
[3 marks total]
**(ii)** Replace integrand by $2 - 2\cos 2\theta$, or equivalent | B1 |
Obtain integral $2\theta - \sin 2\theta$, or equivalent | B1√ |
Substitute limits correctly in an integral of the form $a\theta \pm b\sin 2\theta$, where $ab \neq 0$ | M1 |
Obtain answer $\frac{1}{3}\pi - \frac{\sqrt{3}}{2}$, or exact equivalent | A1 |
[4 marks total]
[The f.t. is on integrands of the form $a + c\cos 2\theta$, where $ac \neq 0$.]5 Let $I = \int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x$.\\
(i) Using the substitution $x = 2 \sin \theta$, show that
$$I = \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 4 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
(ii) Hence find the exact value of $I$.
\hfill \mbox{\textit{CAIE P3 2010 Q5 [7]}}