OCR C4 2012 June — Question 2 5 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2012
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeIntegration of ln(x) alone
DifficultyStandard +0.3 This is a standard integration by parts question requiring students to recognize ln(x+2) as a single function and use the technique with u=ln(x+2) and dv=dx. While it requires knowing the standard trick for integrating logarithms alone, it's a direct application of a well-practiced technique with straightforward algebra and a simple substitution, making it slightly easier than average.
Spec1.08i Integration by parts
2 Use integration by parts to find \(\int \ln ( x + 2 ) \mathrm { d } x\).
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
Write (or imply) \(\int 1\cdot\ln(x+2)\,dx\); \((\ln x + \ln 2 \to\) M0)M1 OR: \(t = \ln(x+2)\) and attempt to connect \(dx\) and \(dt\)
Correct 'by parts' 1st stage: \(x\ln(x+2) - \int\frac{x}{x+2}\,dx\)A1 \(\int te^t\,dt\)
Any suitable starting idea for integrating \(\frac{x}{x+2}\)M1 Attempt by parts with \(u=t\), \(\frac{dv}{dt}=e^t\)
\(\int\frac{x}{x+2}\,dx = x - 2\ln(x+2)\) or \(x+2-2\ln(x+2)\)A1 \(te^t - e^t\)
Overall result \(= x\ln(x+2) - x + 2\ln(x+2)\) [(+c) or (-2+c)]A1
Total: [5]
SR: Correct answer with no working(B2) 
# Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Write (or imply) $\int 1\cdot\ln(x+2)\,dx$; $(\ln x + \ln 2 \to$ M0) | M1 | OR: $t = \ln(x+2)$ and attempt to connect $dx$ and $dt$ |
| Correct 'by parts' 1st stage: $x\ln(x+2) - \int\frac{x}{x+2}\,dx$ | A1 | $\int te^t\,dt$ |
| Any suitable starting idea for integrating $\frac{x}{x+2}$ | M1 | Attempt by parts with $u=t$, $\frac{dv}{dt}=e^t$ |
| $\int\frac{x}{x+2}\,dx = x - 2\ln(x+2)$ or $x+2-2\ln(x+2)$ | A1 | $te^t - e^t$ |
| Overall result $= x\ln(x+2) - x + 2\ln(x+2)$ [(+c) or (-2+c)] | A1 | |
| **Total: [5]** | | |
| SR: Correct answer with no working | (B2) | |

---
2 Use integration by parts to find $\int \ln ( x + 2 ) \mathrm { d } x$.

\hfill \mbox{\textit{OCR C4 2012 Q2 [5]}}
This paper (10 questions)
View full paper
Q1 6 Q2 5 Q3 7 Q4 6 Q5 5 Q6 7 Q7 7 Q8 10 Q9 9 Q10 10