| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.8 This is a multi-part question requiring: (i) simple analysis of ln x minimum, (ii) second derivative calculation with ln x, (iii) non-trivial substitution with exponential transformation and careful limit handling, (iv) integration by parts applied twice. The substitution step requires insight into how limits transform and recognizing the resulting integral form. While each technique is standard for P3, the combination and the need to apply integration by parts twice makes this moderately challenging, above average difficulty. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.08h Integration by substitution1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State at any stage that the x-coordinate of \(A\) is equal to \(1\), or that \(A\) is the point \((1,0)\) | B1 | Total: 1 mark |
| (ii) State \(f'(x) = 2\frac{\ln x}{x}\), or equivalent | B1 | |
| Use product or quotient rule for the next differentiation | M1 | |
| Obtain \(2.\frac{1}{x}.\frac{1}{x} + 2\ln x.\left[\frac{-1}{x^2}\right]\), or any equivalent correct unsimplified form | A1 | |
| Verify that \(f''(e) = 0\) | A1 | Total: 4 marks |
| (iii) State or imply area is \(\int_1^e (\ln x)^2 \, dx\) | B1 | |
| Use \(\frac{dx}{du} = e^u\), or equivalent, in substituting for \(x\) throughout | M1 | |
| Obtain given answer correctly (allow change of limits to be done mentally) | A1 | Total: 3 marks |
| (iv) Attempt the first integration by parts, going the correct way | M1 | |
| Obtain \((u^2 - 2u \pm 2)e^u\), or equivalent, after two applications of the rule | A1 | |
| Obtain exact answer in terms of \(e\), in any correct form, e.g. \((e - 2e + 2) - 2\), or \(e - 2\) | A1 | Total: 3 marks |
| Guidance: The substitution in (iii) may be done in reverse i.e. starting with the \(u\) integral and obtaining the \(x\) integral. The M1A1 scheme applies, but only an explicit statement will earn the B1. | ||
| The M1A1 in (iv) applies to those working in terms of \(x\) and obtaining \(x(ln x)^2 - 2\ln x \pm 2) - 2\ln x \pm 2)\), or equivalent. |
**(i)** State at any stage that the x-coordinate of $A$ is equal to $1$, or that $A$ is the point $(1,0)$ | B1 | **Total: 1 mark**
**(ii)** State $f'(x) = 2\frac{\ln x}{x}$, or equivalent | B1 |
Use product or quotient rule for the next differentiation | M1 |
Obtain $2.\frac{1}{x}.\frac{1}{x} + 2\ln x.\left[\frac{-1}{x^2}\right]$, or any equivalent correct unsimplified form | A1 |
Verify that $f''(e) = 0$ | A1 | **Total: 4 marks**
**(iii)** State or imply area is $\int_1^e (\ln x)^2 \, dx$ | B1 |
Use $\frac{dx}{du} = e^u$, or equivalent, in substituting for $x$ throughout | M1 |
Obtain given answer correctly (allow change of limits to be done mentally) | A1 | **Total: 3 marks**
**(iv)** Attempt the first integration by parts, going the correct way | M1 |
Obtain $(u^2 - 2u \pm 2)e^u$, or equivalent, after two applications of the rule | A1 |
Obtain exact answer in terms of $e$, in any correct form, e.g. $(e - 2e + 2) - 2$, or $e - 2$ | A1 | **Total: 3 marks**
**Guidance:** The substitution in (iii) may be done in reverse i.e. starting with the $u$ integral and obtaining the $x$ integral. The M1A1 scheme applies, but only an explicit statement will earn the B1. | |
The M1A1 in (iv) applies to those working in terms of $x$ and obtaining $x(ln x)^2 - 2\ln x \pm 2) - 2\ln x \pm 2)$, or equivalent. | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{0f081749-4fe0-46e3-96c2-466e69cf49d3-4_620_894_338_687}
The function f is defined by $\mathrm { f } ( x ) = ( \ln x ) ^ { 2 }$ for $x > 0$. The diagram shows a sketch of the graph of $y = \mathrm { f } ( x )$. The minimum point of the graph is $A$. The point $B$ has $x$-coordinate e .\\
(i) State the $x$-coordinate of $A$.\\
(ii) Show that $\mathrm { f } ^ { \prime \prime } ( x ) = 0$ at $B$.\\
(iii) Use the substitution $x = \mathrm { e } ^ { u }$ to show that the area of the region bounded by the $x$-axis, the line $x = \mathrm { e }$, and the part of the curve between $A$ and $B$ is given by
$$\int _ { 0 } ^ { 1 } u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u .$$
(iv) Hence, or otherwise, find the exact value of this area.
\hfill \mbox{\textit{CAIE P3 2002 Q10 [11]}}