OCR MEI C3 — Question 5 4 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks4
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Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeShow definite integral equals specific value (algebraic/exponential substitution)
DifficultyModerate -0.3 This is a straightforward integration question requiring recognition that the numerator is (up to a constant) the derivative of the denominator, leading to a logarithmic result. The definite integral evaluation is mechanical once the substitution u = x² + 2 is identified. Slightly easier than average due to the standard pattern and simple arithmetic.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
Show that \(\int_1^4 \frac{x}{x^2 + 2} \, dx = \frac{1}{2} \ln 6\). [4]
AnswerMarks Guidance
Answer/Working: Let \(u = x^2 + 2 \Rightarrow du = 2x dx\); \(\int_1^3 \frac{x}{x^2+2}dx = \int_3^{11/2} \frac{1}{u} du = \frac{1}{2}[\ln u]_3^{11/2}\)M1, A1 \(\int 1/u^2 du\) or \(\ln(x^2+1)\); \(\frac{1}{2}\ln u\) or \(\frac{1}{2}\ln(x^2+2)\)
Answer/Working: \(= \frac{1}{2}(\ln 18 - \ln 3) = \frac{1}{2}\ln(18/3) = \frac{1}{2}\ln 6^*\)M1, E1, [4] substituting correct limits (\(u\) or \(x\)); must show working for ln 6 
**Answer/Working:** Let $u = x^2 + 2 \Rightarrow du = 2x dx$; $\int_1^3 \frac{x}{x^2+2}dx = \int_3^{11/2} \frac{1}{u} du = \frac{1}{2}[\ln u]_3^{11/2}$ | **M1, A1** | $\int 1/u^2 du$ or $\ln(x^2+1)$; $\frac{1}{2}\ln u$ or $\frac{1}{2}\ln(x^2+2)$

**Answer/Working:** $= \frac{1}{2}(\ln 18 - \ln 3) = \frac{1}{2}\ln(18/3) = \frac{1}{2}\ln 6^*$ | **M1, E1, [4]** | substituting correct limits ($u$ or $x$); must show working for ln 6
Show that $\int_1^4 \frac{x}{x^2 + 2} \, dx = \frac{1}{2} \ln 6$. [4]

\hfill \mbox{\textit{OCR MEI C3  Q5 [4]}}
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Q1 5 Q2 18 Q3 19 Q4 4 Q5 4