| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show integral transforms via substitution then evaluate (trigonometric/Weierstrass) |
| Difficulty | Standard +0.3 This is a guided substitution problem where part (i) walks students through the algebraic manipulation (finding dx, simplifying the integrand), and part (ii) requires applying the double angle formula and evaluating definite integral limits. While it involves multiple steps, the substitution is given explicitly and the techniques (double angle identity, basic integration) are standard P3 material with no novel insight required. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08h Integration by substitution |
| Answer | Marks |
|---|---|
| State \(\frac{dx}{d\theta} = 2\sin \theta \cos \theta\), or \(dx = 2\sin \theta \cos \theta d\theta\) | B1 |
| Substitute for \(x\) and \(dx\) throughout | M1 |
| Obtain any correct form in terms of \(\theta\) | A1 |
| Reduce to the given form correctly | A1 |
| Answer | Marks |
|---|---|
| Use \(\cos 2\lambda\) formula, replacing integrand by \(a + b\cos 2\theta\), where \(ab \neq 0\) | M1* |
| Integrate and obtain \(\theta - \frac{1}{2}\sin 2\theta\) | A1√ |
| Use limits \(\theta = 0\) and \(\theta = \frac{1}{6}\pi\) | M1(dep*) |
| Obtain exact answer \(\frac{1}{6}\pi - \frac{1}{4}\sqrt{3}\), or equivalent | A1 |
**(i)**
State $\frac{dx}{d\theta} = 2\sin \theta \cos \theta$, or $dx = 2\sin \theta \cos \theta d\theta$ | B1 |
Substitute for $x$ and $dx$ throughout | M1 |
Obtain any correct form in terms of $\theta$ | A1 |
Reduce to the given form correctly | A1 |
**Total for (i): [4]**
**(ii)**
Use $\cos 2\lambda$ formula, replacing integrand by $a + b\cos 2\theta$, where $ab \neq 0$ | M1* |
Integrate and obtain $\theta - \frac{1}{2}\sin 2\theta$ | A1√ |
Use limits $\theta = 0$ and $\theta = \frac{1}{6}\pi$ | M1(dep*) |
Obtain exact answer $\frac{1}{6}\pi - \frac{1}{4}\sqrt{3}$, or equivalent | A1 |
**Total for (ii): [4]**
---6 (i) Use the substitution $x = \sin ^ { 2 } \theta$ to show that
$$\int \sqrt { } \left( \frac { x } { 1 - x } \right) \mathrm { d } x = \int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$$
(ii) Hence find the exact value of
$$\left. \int _ { 0 } ^ { \frac { 1 } { 4 } } \sqrt { ( } \frac { x } { 1 - x } \right) \mathrm { d } x$$
\hfill \mbox{\textit{CAIE P3 2005 Q6 [8]}}