OCR MEI C3 — Question 6 6 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeIntegration of ln(x) alone
DifficultyStandard +0.3 This is a guided integration by parts question where students are explicitly told to differentiate x ln(x) first, which directly reveals the integration technique needed. While integration by parts with ln(x) is a standard C3 topic, the scaffolding makes this easier than average. The algebraic manipulation to reach the final form ln(a) + b is straightforward, making this slightly above routine but not challenging.
Spec1.07l Derivative of ln(x): and related functions1.08i Integration by parts
6 Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x )\) and hence or otherwise find the value of \(\int _ { 2 } ^ { 3 } \ln x \mathrm {~d} x\), giving your answer in the form \(\ln a + b\), where \(a\) and \(b\) are to be determined.
Question 6:
AnswerMarks Guidance
AnswerMark Guidance
\(\frac{d}{dx}(x\ln x) = \ln x + x \times \frac{1}{x} = \ln x + 1\)M1, A1 Product
\(\Rightarrow x\ln x = \int(\ln x + 1)\,dx = \int\ln x\,dx + x\)M1, A1
\(\Rightarrow \int_2^3 \ln x\,dx = \left[x\ln x - x\right]_2^3 = (3\ln 3 - 3) - (2\ln 2 - 2)\)M1 Integrand, limits
\(= 3\ln 3 - 2\ln 2 - 1 = \ln\frac{27}{4} - 1\)A1
Total: 6 
## Question 6:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{d}{dx}(x\ln x) = \ln x + x \times \frac{1}{x} = \ln x + 1$ | M1, A1 | Product |
| $\Rightarrow x\ln x = \int(\ln x + 1)\,dx = \int\ln x\,dx + x$ | M1, A1 | |
| $\Rightarrow \int_2^3 \ln x\,dx = \left[x\ln x - x\right]_2^3 = (3\ln 3 - 3) - (2\ln 2 - 2)$ | M1 | Integrand, limits |
| $= 3\ln 3 - 2\ln 2 - 1 = \ln\frac{27}{4} - 1$ | A1 | |
| **Total: 6** | | |

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6 Find $\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x )$ and hence or otherwise find the value of $\int _ { 2 } ^ { 3 } \ln x \mathrm {~d} x$, giving your answer in the form $\ln a + b$, where $a$ and $b$ are to be determined.

\hfill \mbox{\textit{OCR MEI C3  Q6 [6]}}
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