| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a straightforward application of substitution u=√x with given limits, requiring students to find du/dx, change limits, and integrate. The stationary point in part (a) provides scaffolding. While it involves logarithmic integration, the substitution is explicitly provided and the method is standard for P3/C4 level, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = 3\) | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\frac{\mathrm{d}u}{\mathrm{d}x} = \frac{1}{2\sqrt{x}}\), or \(\mathrm{d}x = 2\sqrt{x}\,\mathrm{d}u\), or \(2u\,\mathrm{d}u = \mathrm{d}x\) | B1 | |
| Substitute and obtain integrand \(\frac{2}{9-u^2}\) | B1 | |
| Use given formula for the integral or integrate relevant partial fractions | M1 | |
| Obtain integral \(\frac{1}{3}\ln\!\left(\frac{3+u}{3-u}\right)\) | A1 | OE |
| Use limits \(u = 0\) and \(u = 2\) correctly | M1 | |
| Obtain the given answer correctly | A1 | |
| Total | 6 |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain answer $x = 3$ | A1 | |
| **Total** | **4** | |
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\frac{\mathrm{d}u}{\mathrm{d}x} = \frac{1}{2\sqrt{x}}$, or $\mathrm{d}x = 2\sqrt{x}\,\mathrm{d}u$, or $2u\,\mathrm{d}u = \mathrm{d}x$ | B1 | |
| Substitute and obtain integrand $\frac{2}{9-u^2}$ | B1 | |
| Use given formula for the integral or integrate relevant partial fractions | M1 | |
| Obtain integral $\frac{1}{3}\ln\!\left(\frac{3+u}{3-u}\right)$ | A1 | OE |
| Use limits $u = 0$ and $u = 2$ correctly | M1 | |
| Obtain the given answer correctly | A1 | |
| **Total** | **6** | |\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of the stationary point of the curve with equation $y = \mathrm { f } ( x )$.
\item Using the substitution $u = \sqrt { x }$, show that $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 3 } \ln 5$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q9 [10]}}