| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (trigonometric substitution or identity) |
| Difficulty | Standard +0.8 This question requires proving a non-standard trigonometric identity (not immediately obvious), then applying it with substitution u=2θ to evaluate a definite integral to a specific logarithmic form. The identity proof requires manipulation of tan 2θ using double angle formulas, and the integration step needs careful handling of limits and recognition that the result simplifies to the given exact value. More demanding than routine integration by substitution due to the proof component and non-standard setup. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae1.05p Proof involving trig: functions and identities1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use tan \(2A\) formula to express LHS in terms of \(\tan\theta\) | M1 | |
| Express as a single fraction in any correct form | A1 | |
| Use Pythagoras or cos \(2A\) formula | M1 | |
| Obtain the given result correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express LHS in terms of \(\sin 2\theta\), \(\cos 2\theta\), \(\sin\theta\) and \(\cos\theta\) | M1 | |
| Express as a single fraction in any correct form | A1 | |
| Use Pythagoras or cos \(2A\) formula or \(\sin(A-B)\) formula | M1 | |
| Obtain the given result correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate and obtain a term of the form \(a\ln(\cos 2\theta)\) or \(b\ln(\cos\theta)\) (or secant equivalents) | M1* | |
| Obtain integral \(-\frac{1}{2}\ln(\cos 2\theta) + \ln(\cos\theta)\), or equivalent | A1 | |
| Substitute limits correctly (expect to see use of both limits) | DM1 | |
| Obtain the given answer following full and correct working | A1 |
## Question 5:
### Part (i):
#### EITHER:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use tan $2A$ formula to express LHS in terms of $\tan\theta$ | M1 | |
| Express as a single fraction in any correct form | A1 | |
| Use Pythagoras or cos $2A$ formula | M1 | |
| Obtain the given result correctly | A1 | |
#### OR:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express LHS in terms of $\sin 2\theta$, $\cos 2\theta$, $\sin\theta$ and $\cos\theta$ | M1 | |
| Express as a single fraction in any correct form | A1 | |
| Use Pythagoras or cos $2A$ formula or $\sin(A-B)$ formula | M1 | |
| Obtain the given result correctly | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate and obtain a term of the form $a\ln(\cos 2\theta)$ or $b\ln(\cos\theta)$ (or secant equivalents) | M1* | |
| Obtain integral $-\frac{1}{2}\ln(\cos 2\theta) + \ln(\cos\theta)$, or equivalent | A1 | |
| Substitute limits correctly (expect to see use of **both** limits) | DM1 | |
| Obtain the given answer following full and correct working | A1 | |
---5 (i) Prove the identity $\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta$.\\
(ii) Hence show that $\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }$.
\hfill \mbox{\textit{CAIE P3 2016 Q5 [8]}}