OCR MEI C3 — Question 5 5 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeBasic integration by parts
DifficultyModerate -0.3 This is a straightforward single application of integration by parts with standard functions (polynomial × exponential), followed by definite integral evaluation. It's slightly easier than average because it's a direct textbook-style question requiring only one technique with no complications, though it does require careful execution of the method and numerical evaluation.
Spec1.08i Integration by parts
5 Find \(\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\), giving your answer to 1 decimal place.
Question 5:
AnswerMarks Guidance
AnswerMark Guidance
\(\int_2^3 xe^{2x}\,dx\), \(u = x\), \(\frac{dv}{dx} = e^{2x}\)M1 Choice of \(u\)
\(\frac{du}{dx} = 1\), \(v = \frac{1}{2}e^{2x}\)A1
\(= \left[\frac{1}{2}xe^{2x}\right]_2^3 - \frac{1}{2}\int_2^3 e^{2x}\,dx = \left[\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}\right]_2^3\)M1, A1, A1
\(= \frac{5}{4}e^6 - \frac{3}{4}e^4 = 463.3\)
Total: 5 
## Question 5:

| Answer | Mark | Guidance |
|--------|------|----------|
| $\int_2^3 xe^{2x}\,dx$, $u = x$, $\frac{dv}{dx} = e^{2x}$ | M1 | Choice of $u$ |
| $\frac{du}{dx} = 1$, $v = \frac{1}{2}e^{2x}$ | A1 | |
| $= \left[\frac{1}{2}xe^{2x}\right]_2^3 - \frac{1}{2}\int_2^3 e^{2x}\,dx = \left[\frac{1}{2}xe^{2x} - \frac{1}{4}e^{2x}\right]_2^3$ | M1, A1, A1 | |
| $= \frac{5}{4}e^6 - \frac{3}{4}e^4 = 463.3$ | | |
| **Total: 5** | | |

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5 Find $\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x$, giving your answer to 1 decimal place.

\hfill \mbox{\textit{OCR MEI C3  Q5 [5]}}
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Q1 2 Q2 3 Q3 7 Q4 7 Q5 5 Q6 6 Q7 6 Q8 18 Q9 18