OCR C4 2007 June — Question 2 6 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2007
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeDouble integration by parts
DifficultyStandard +0.3 This is a standard double integration by parts question with straightforward choices (u = x², dv = e^x dx, then u = x, dv = e^x dx). The limits are simple (0 and 1), making evaluation clean. While it requires two applications of the technique, this is a textbook exercise with no conceptual challenges beyond executing the standard algorithm correctly.
Spec1.08i Integration by parts
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
AnswerMarks Guidance
Use parts with \(u = x^2, dv = e^x\)*M1 obtaining a result \(f(x) +/- \int g(x)(dx)\)
Obtain \(x^2e^x - \int 2xe^x (dx)\)A1
Attempt parts again with \(u = (-)(2)x, dv = e^x\)M1
Final is \((x^2 - 2x + 2)e^x\) AEF incl bracketsA1 s.o.i. eg \(e + (-2x + 2)e^x\)
Use limits correctly throughoutdep*M1 Tolerate (their value for \(x = 1\)) \((-0)\)
\(e^{(1)} - 2\) ISW Exact answer onlyA1 Allow \(0.718 \to M1\) 
Use parts with $u = x^2, dv = e^x$ | *M1 | obtaining a result $f(x) +/- \int g(x)(dx)$
Obtain $x^2e^x - \int 2xe^x (dx)$ | A1 |
Attempt parts again with $u = (-)(2)x, dv = e^x$ | M1 |
Final is $(x^2 - 2x + 2)e^x$ AEF incl brackets | A1 | s.o.i. eg $e + (-2x + 2)e^x$
Use limits correctly throughout | dep*M1 | Tolerate (their value for $x = 1$) $(-0)$
$e^{(1)} - 2$ ISW Exact answer only | A1 | Allow $0.718 \to M1$ | 6

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2 Find the exact value of $\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x$.

\hfill \mbox{\textit{OCR C4 2007 Q2 [6]}}
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