Standard +0.3 This is a straightforward integration by substitution problem with standard algebraic manipulation. The substitution is given explicitly, and students need to rewrite the integrand, change limits, integrate (which splits into two simple terms), and solve for c using logarithm properties. While it requires careful execution across multiple steps, each individual step is routine for A-level Further Maths students.
5 In this question you must show detailed reasoning.
Using the substitution \(\mathrm { u } = \mathrm { x } + 1\), find the value of the positive integer \(c\) such that \(\int _ { \mathrm { c } } ^ { \mathrm { c } + 4 } \frac { \mathrm { x } } { ( \mathrm { x } + 1 ) ^ { 2 } } \mathrm { dx } = \ln 3 - \frac { 1 } { 3 }\).
Absence of \(du\) allowed; sight of \(dx\) loses mark
Integrating to get \(\ln u\) term OR power of \(u\) term
M1
Must attempt to integrate *their* expression
Correct answer e.g. \(\ln(c+4+1)\); allow unsimplified
A1
Dependent on three M marks; condone missing brackets if recovered
Setting up equation using *their* \(\ln\) fraction \(= 3\) OR *their* other fraction(s) \(= \frac{-1}{3}\)
M1
May be unsimplified
\(c = 1\) AND checking in other equation
A1
Must verify \(c=1\) in second equation
*Note: Integration by parts loses first three M marks. Award SC1 for correct integral; final M1A1 still available.*
## Question 5:
| $du = dx$ or equivalent (e.g. $du/dx = 1$) | M1 | First substitution step |
| Forming the integral | M1 | Absence of $du$ allowed; sight of $dx$ loses mark |
| Integrating to get $\ln u$ term OR power of $u$ term | M1 | Must attempt to integrate *their* expression |
| Correct answer e.g. $\ln(c+4+1)$; allow unsimplified | A1 | Dependent on three M marks; condone missing brackets if recovered |
| Setting up equation using *their* $\ln$ fraction $= 3$ OR *their* other fraction(s) $= \frac{-1}{3}$ | M1 | May be unsimplified |
| $c = 1$ AND checking in other equation | A1 | Must verify $c=1$ in second equation |
*Note: Integration by parts loses first three M marks. Award SC1 for correct integral; final M1A1 still available.*
5 In this question you must show detailed reasoning.
Using the substitution $\mathrm { u } = \mathrm { x } + 1$, find the value of the positive integer $c$ such that $\int _ { \mathrm { c } } ^ { \mathrm { c } + 4 } \frac { \mathrm { x } } { ( \mathrm { x } + 1 ) ^ { 2 } } \mathrm { dx } = \ln 3 - \frac { 1 } { 3 }$.
\hfill \mbox{\textit{OCR MEI Paper 3 2024 Q5 [6]}}