| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Partial fractions after substitution |
| Difficulty | Standard +0.3 This is a straightforward C4 integration question combining standard substitution with partial fractions. Part (i) is routine verification of a given substitution (finding du/dx and substituting), while part (ii) requires standard partial fractions decomposition followed by logarithmic integration. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt to connect \(du\) and \(dx\), find \(\frac{du}{dx}\) or \(\frac{dx}{du}\) | M1 | But not e.g. \(du = dx\) |
| Any correct relationship, e.g. \(dx = 2u\, du\) | A1 | or \(\frac{du}{dx} = \frac{1}{2}x^{-\frac{1}{2}}\) |
| Subst with clear reduction (\(\geq 1\) inter step) to AG | A1 (3) | WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt partial fractions | M1 | |
| \(\frac{2}{u} - \frac{2}{1+u}\) | A1 | |
| \(\sqrt{\ } A\ln u + B\ln(1+u)\) | \(\sqrt{}\)A1 | Based on \(\frac{A}{u}+\frac{B}{1+u}\) |
| Attempt integ, change limits & use on \(f(u)\) | M1 | or re-subst & use 1 & 9 |
| \(\ln\frac{9}{4}\) AEexactF (e.g. \(2\ln 3 - 2\ln 4 + 2\ln 2\)) | A1 (5) | Not involving \(\ln 1\) |
| Total: 8 |
# Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to connect $du$ and $dx$, find $\frac{du}{dx}$ or $\frac{dx}{du}$ | M1 | But not e.g. $du = dx$ |
| Any correct relationship, e.g. $dx = 2u\, du$ | A1 | or $\frac{du}{dx} = \frac{1}{2}x^{-\frac{1}{2}}$ |
| Subst with clear reduction ($\geq 1$ inter step) to **AG** | A1 **(3)** | WWW |
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# Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt partial fractions | M1 | |
| $\frac{2}{u} - \frac{2}{1+u}$ | A1 | |
| $\sqrt{\ } A\ln u + B\ln(1+u)$ | $\sqrt{}$A1 | Based on $\frac{A}{u}+\frac{B}{1+u}$ |
| Attempt integ, change limits & use on $f(u)$ | M1 | or re-subst & use 1 & 9 |
| $\ln\frac{9}{4}$ AEexactF (e.g. $2\ln 3 - 2\ln 4 + 2\ln 2$) | A1 **(5)** | Not involving $\ln 1$ |
| **Total: 8** | | |
---5 (i) Show that the substitution $u = \sqrt { x }$ transforms $\int \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x$ to $\int \frac { 2 } { u ( 1 + u ) } \mathrm { d } u$.\\
(ii) Hence find the exact value of $\int _ { 1 } ^ { 9 } \frac { 1 } { x ( 1 + \sqrt { x } ) } \mathrm { d } x$.
\hfill \mbox{\textit{OCR C4 2009 Q5 [8]}}