| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Stationary points then area/volume |
| Difficulty | Standard +0.3 Part (i) requires standard differentiation using product rule and solving for stationary points. Part (ii) is a straightforward application of integration by parts with u=ln x, though students must handle the limits carefully. Both parts are routine techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Use product rule | M1 | |
| Obtain derivative in any correct form e.g. \(\frac{x^{\frac{1}{2}}}{2} + x^{-1}\ln x\) | A1 | |
| Equate derivative to zero and solve for \(\ln x\) | M1 | |
| Obtain \(x = e^{-2}\) (or \(\frac{1}{e^2}\)) or equivalent | A1 | 4 |
| Answer | Marks |
|---|---|
| Attempt integration by parts with \(u = \ln x\) | M1 |
| Obtain \(\frac{2}{3}x^{\frac{3}{2}}\ln x - \int\frac{2}{3}y^{\frac{1}{2}}_{\frac{1}{x}}dx\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt integration by parts with \(u = x^{\frac{1}{2}}\) | M1 | |
| Obtain \(x^{\frac{1}{2}}(x\ln x - x) - \int(x\ln x - x) \cdot \frac{x^{-1}}{2}dx\) | A1 | |
| Obtain indefinite integral \(\frac{2}{3}x^{\frac{3}{2}}\ln x - \frac{4}{9}x^{\frac{3}{2}}\), or equivalent | A1 | |
| Use \(x = 1\) and \(x = 4\) as limits | M1 | |
| Obtain answer \(4.28\) | A1 | 5 |
**(i)**
| Use product rule | M1 |
| Obtain derivative in any correct form e.g. $\frac{x^{\frac{1}{2}}}{2} + x^{-1}\ln x$ | A1 |
| Equate derivative to zero and solve for $\ln x$ | M1 |
| Obtain $x = e^{-2}$ (or $\frac{1}{e^2}$) or equivalent | A1 | 4 |
**(ii) EITHER:**
| Attempt integration by parts with $u = \ln x$ | M1 |
| Obtain $\frac{2}{3}x^{\frac{3}{2}}\ln x - \int\frac{2}{3}y^{\frac{1}{2}}_{\frac{1}{x}}dx$, or equivalent | A1 |
**OR:**
| Attempt integration by parts with $u = x^{\frac{1}{2}}$ | M1 |
| Obtain $x^{\frac{1}{2}}(x\ln x - x) - \int(x\ln x - x) \cdot \frac{x^{-1}}{2}dx$ | A1 |
| Obtain indefinite integral $\frac{2}{3}x^{\frac{3}{2}}\ln x - \frac{4}{9}x^{\frac{3}{2}}$, or equivalent | A1 |
| Use $x = 1$ and $x = 4$ as limits | M1 |
| Obtain answer $4.28$ | A1 | 5 |\includegraphics{figure_8}
The diagram shows a sketch of the curve $y = x^2\ln x$ and its minimum point $M$. The curve cuts the $x$-axis at the point $(1, 0)$.
\begin{enumerate}[label=(\roman*)]
\item Find the exact value of the $x$-coordinate of $M$. [4]
\item Use integration by parts to find the area of the shaded region enclosed by the curve, the $x$-axis and the line $x = 4$. Give your answer correct to 2 decimal places. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2006 Q8 [9]}}