| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Integration involving inverse trig |
| Difficulty | Standard +0.8 Part (a) is straightforward implicit differentiation requiring knowledge of d/dy(tan y). Part (b) requires recognizing that the result from (a) enables integration by parts with u = tan^(-1)(2x), combining multiple techniques (IBP, substitution, and exact evaluation) in a non-routine way. The inverse trig context and exact value requirement elevate this above standard IBP questions. |
| Spec | 1.07s Parametric and implicit differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(2 = \sec^2 y \dfrac{dy}{dx}\) or equivalent | B1 | E.g. \(2\dfrac{dx}{dy} = \sec^2 y\) by differentiation with respect to \(y\) |
| Use \(\sec^2 y = 1 + \tan^2 y\) | M1 | |
| Replace \(\tan y\) with \(2x\) and rearrange to obtain \(\dfrac{dy}{dx} = \dfrac{2}{1+4x^2}\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate by parts and reach \(ax^2\tan^{-1}2x + b\displaystyle\int \dfrac{x^2}{1+4x^2}\,dx\) | \*M1 | |
| Obtain \(\dfrac{1}{2}x^2\tan^{-1}2x - \displaystyle\int\dfrac{x^2}{1+4x^2}\,dx\) | A1 | OE |
| Reduce integral to expression of the form \(\displaystyle\int m + \dfrac{n}{1+4x^2}\,dx\) | M1 | |
| Complete integration and reach \(px^2\tan^{-1}2x + qx + r\tan^{-1}2x\) | M1 | |
| Obtain \(\dfrac{1}{2}x^2\tan^{-1}2x - \dfrac{1}{4}x + \dfrac{1}{8}\tan^{-1}2x\) | A1 | OE |
| Use limits \(x = \dfrac{1}{2}\) and \(x = \dfrac{1}{2}\sqrt{3}\) in correct order, having integrated twice | DM1 | |
| Obtain \(\dfrac{5}{48}\pi - \dfrac{1}{8}\sqrt{3} + \dfrac{1}{8}\) or exact equivalent | A1 | |
| 7 |
## Question 10:
**Part 10(a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $2 = \sec^2 y \dfrac{dy}{dx}$ or equivalent | B1 | E.g. $2\dfrac{dx}{dy} = \sec^2 y$ by differentiation with respect to $y$ |
| Use $\sec^2 y = 1 + \tan^2 y$ | M1 | |
| Replace $\tan y$ with $2x$ and rearrange to obtain $\dfrac{dy}{dx} = \dfrac{2}{1+4x^2}$ | A1 | |
| | **3** | |
**Part 10(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $ax^2\tan^{-1}2x + b\displaystyle\int \dfrac{x^2}{1+4x^2}\,dx$ | \*M1 | |
| Obtain $\dfrac{1}{2}x^2\tan^{-1}2x - \displaystyle\int\dfrac{x^2}{1+4x^2}\,dx$ | A1 | OE |
| Reduce integral to expression of the form $\displaystyle\int m + \dfrac{n}{1+4x^2}\,dx$ | M1 | |
| Complete integration and reach $px^2\tan^{-1}2x + qx + r\tan^{-1}2x$ | M1 | |
| Obtain $\dfrac{1}{2}x^2\tan^{-1}2x - \dfrac{1}{4}x + \dfrac{1}{8}\tan^{-1}2x$ | A1 | OE |
| Use limits $x = \dfrac{1}{2}$ and $x = \dfrac{1}{2}\sqrt{3}$ in correct order, having integrated twice | DM1 | |
| Obtain $\dfrac{5}{48}\pi - \dfrac{1}{8}\sqrt{3} + \dfrac{1}{8}$ or exact equivalent | A1 | |
| | **7** | |
---10
\begin{enumerate}[label=(\alph*)]
\item Given that $2 x = \tan y$, show that $\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }$.
\item Hence find the exact value of $\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q10 [10]}}