Edexcel FP3 2010 June — Question 2 5 marks

Exam BoardEdexcel
ModuleFP3 (Further Pure Mathematics 3)
Year2010
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicIntegration using inverse trig and hyperbolic functions
TypeStandard integral of 1/(a²+x²)
DifficultyStandard +0.3 This is a standard FP3 integration question requiring completing the square to get the form 1/(a²+u²), then applying the arctan formula. While it's a Further Maths topic (making it harder than typical A-level), it's a routine textbook exercise with a clear method and no novel insight required, placing it slightly above average difficulty overall.
Spec1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.08h Integration by substitution
2. Use calculus to find the exact value of \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\).
Question 2:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(x^2 + 4x + 13 = (x+2)^2 + 9\)B1 Correct completion of the square
\(\int \frac{1}{(x+2)^2+9}dx = \frac{1}{3}\arctan\left(\frac{x+2}{3}\right)\)M1 A1 M1 for arctan form, A1 correct
\(\left[\frac{1}{3}\arctan\left(\frac{x+2}{3}\right)\right]_{-2}^{1} = \frac{1}{3}(\arctan 1 - \arctan 0)\)M1 Substituting limits
\(= \frac{\pi}{12}\)A1 (5) 
## Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^2 + 4x + 13 = (x+2)^2 + 9$ | B1 | Correct completion of the square |
| $\int \frac{1}{(x+2)^2+9}dx = \frac{1}{3}\arctan\left(\frac{x+2}{3}\right)$ | M1 A1 | M1 for arctan form, A1 correct |
| $\left[\frac{1}{3}\arctan\left(\frac{x+2}{3}\right)\right]_{-2}^{1} = \frac{1}{3}(\arctan 1 - \arctan 0)$ | M1 | Substituting limits |
| $= \frac{\pi}{12}$ | A1 | (5) |

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2. Use calculus to find the exact value of $\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x$.\\

\hfill \mbox{\textit{Edexcel FP3 2010 Q2 [5]}}
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Q1 5 Q2 5 Q3 8 Q4 8 Q5 9 Q6 13 Q7 14 Q8 13