| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| 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 question requires executing a trigonometric substitution with careful handling of limits, simplifying the resulting trigonometric expression, then integrating cos²θ using double angle formulas. While the substitution is guided and the techniques are standard for Further Maths Pure 3, the multi-step algebraic manipulation and exact evaluation make it moderately challenging, placing it 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 |
|---|---|---|
| (i) State or imply \(dx = \sqrt{3}\sec^2\theta \, d\theta\) | B1 | |
| Substitute for \(x\) and \(dx\) throughout | M1 | |
| Obtain the given answer correctly | A1 | [3] |
| (ii) Replace integrand by \(\frac{1}{2}\cos 2\theta + \frac{1}{2}\) | B1 | |
| Obtain integral \(\frac{1}{4}\sin 2\theta + \frac{1}{4}\theta\) | B1* | |
| Substitute limits correctly in an integral of the form \(c\sin 2\theta + b\theta\), where \(cb \neq 0\) | M1 | |
| Obtain answer \(\frac{1}{12}\sqrt{3\pi} + \frac{3}{4}\), or exact equivalent | A1 | [4] |
(i) State or imply $dx = \sqrt{3}\sec^2\theta \, d\theta$ | B1 |
Substitute for $x$ and $dx$ throughout | M1 |
Obtain the given answer correctly | A1 | [3]
(ii) Replace integrand by $\frac{1}{2}\cos 2\theta + \frac{1}{2}$ | B1 |
Obtain integral $\frac{1}{4}\sin 2\theta + \frac{1}{4}\theta$ | B1* |
Substitute limits correctly in an integral of the form $c\sin 2\theta + b\theta$, where $cb \neq 0$ | M1 |
Obtain answer $\frac{1}{12}\sqrt{3\pi} + \frac{3}{4}$, or exact equivalent | A1 | [4] |
[The f.t. is on integrands of the form $a\cos 2\theta + b$, where $ab \neq 0$.]5 Let $I = \int _ { 0 } ^ { 1 } \frac { 9 } { \left( 3 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$.\\
(i) Using the substitution $x = ( \sqrt { } 3 ) \tan \theta$, show that $I = \sqrt { } 3 \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta$.\\
(ii) Hence find the exact value of $I$.
\hfill \mbox{\textit{CAIE P3 2016 Q5 [7]}}