OCR C4 2011 January — Question 7 7 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Year2011
SessionJanuary
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeShow that integral equals expression
DifficultyStandard +0.8 This requires applying integration by parts twice (once for x² term, once for x term) with careful bookkeeping of constants, plus evaluating at limits involving π. While the technique is standard C4 material, the repeated application and algebraic manipulation of multiple terms makes it more demanding than a single-application question, placing it moderately above average difficulty.
Spec1.08i Integration by parts
7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).
AnswerMarks Guidance
Attempt parts with \(u = x^2 + 5x + 7\), \(dv = \sin x\)M1 as far as \(f(x) + /-\int g(x)dx\)
1st stage \(= -(x^2 + 5x + 7)\cos x + \int (2x+5)\cos x\,dx\)A1 signs need not be amalgamated at this stage
\(\int(2x+5)\cos x\,dx = (2x+5)\sin x - \int 2\sin x\,dx\)B1 indep of previous A1 being awarded
\(= (2x+5)\sin x + 2\cos x\)B1
\(I = -(x^2 + 5x + 7)\cos x + (2x+5)\sin x + 2\cos x\)A1 WWW
(Substitute \(x = \pi\)) – (Substitute \(x = 0\))M1 An attempt at subst \(x = 0\) must be seen
\(\pi^2 + 5\pi + 10\) WWW AGA1 7 
Attempt parts with $u = x^2 + 5x + 7$, $dv = \sin x$ | M1 | as far as $f(x) + /-\int g(x)dx$
1st stage $= -(x^2 + 5x + 7)\cos x + \int (2x+5)\cos x\,dx$ | A1 | signs need not be amalgamated at this stage
$\int(2x+5)\cos x\,dx = (2x+5)\sin x - \int 2\sin x\,dx$ | B1 | indep of previous A1 being awarded
$= (2x+5)\sin x + 2\cos x$ | B1 |
$I = -(x^2 + 5x + 7)\cos x + (2x+5)\sin x + 2\cos x$ | A1 | WWW
(Substitute $x = \pi$) – (Substitute $x = 0$) | M1 | An attempt at subst $x = 0$ must be seen
$\pi^2 + 5\pi + 10$ WWW AG | A1 7 |

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7 Show that $\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10$.

\hfill \mbox{\textit{OCR C4 2011 Q7 [7]}}
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Q1 6 Q2 7 Q3 8 Q4 7 Q5 9 Q6 10 Q7 7 Q8 8 Q9 10