| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Show definite integral equals specific value (requiring partial fractions or complex algebra) |
| Difficulty | Standard +0.3 This is a standard integration by substitution question with a straightforward substitution (u = √x) followed by partial fractions. The substitution mechanics are routine (finding du/dx and changing limits), and the partial fractions decomposition of 2/[u(4-u)] is textbook. While it requires multiple steps, each step follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(du = \frac{1}{2\sqrt{x}}dx\), or \(2u\,du = dx\), or \(\frac{du}{dx} = \frac{1}{2\sqrt{x}}\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\) throughout the integral | M1 | |
| Obtain the given form of indefinite integral correctly with no errors seen | A1 | Total: 3 marks |
| (ii) Attempting to express the integrand as \(\frac{A}{u} + \frac{B}{4-u}\), use a correct method to find either \(A\) or \(B\) | M1* | |
| Obtain \(A = \frac{1}{2}\) and \(B = \frac{1}{2}\) | A1 | |
| Integrate and obtain \(\frac{1}{2}\ln u - \frac{1}{2}\ln(4-u)\), or equivalent | A1 \(\checkmark\) + A1 \(\checkmark\) | |
| Use limits \(u = 1\) and \(u = 2\) correctly, or equivalent, in an integral of the form \(c\ln u + d\ln(4-u)\) | M1(dep*) | |
| Obtain given answer correctly following full and exact working | A1 | Total: 6 marks |
(i) State or imply $du = \frac{1}{2\sqrt{x}}dx$, or $2u\,du = dx$, or $\frac{du}{dx} = \frac{1}{2\sqrt{x}}$, or equivalent | B1 |
Substitute for $x$ and $dx$ throughout the integral | M1 |
Obtain the given form of indefinite integral correctly with no errors seen | A1 | **Total: 3 marks**
(ii) Attempting to express the integrand as $\frac{A}{u} + \frac{B}{4-u}$, use a correct method to find either $A$ or $B$ | M1* |
Obtain $A = \frac{1}{2}$ and $B = \frac{1}{2}$ | A1 |
Integrate and obtain $\frac{1}{2}\ln u - \frac{1}{2}\ln(4-u)$, or equivalent | A1 $\checkmark$ + A1 $\checkmark$ |
Use limits $u = 1$ and $u = 2$ correctly, or equivalent, in an integral of the form $c\ln u + d\ln(4-u)$ | M1(dep*) |
Obtain given answer correctly following full and exact working | A1 | **Total: 6 marks**
---7 Let $I = \int _ { 1 } ^ { 4 } \frac { 1 } { x ( 4 - \sqrt { } x ) } \mathrm { d } x$.\\
(i) Use the substitution $u = \sqrt { } x$ to show that $I = \int _ { 1 } ^ { 2 } \frac { 2 } { u ( 4 - u ) } \mathrm { d } u$.\\
(ii) Hence show that $I = \frac { 1 } { 2 } \ln 3$.
\hfill \mbox{\textit{CAIE P3 2007 Q7 [9]}}