Edexcel C2 2006 June — Question 2 5 marks

Exam BoardEdexcel
ModuleC2 (Core Mathematics 2)
Year2006
SessionJune
Marks5
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TopicStandard Integrals and Reverse Chain Rule
TypeFind indefinite integral of polynomial/power
DifficultyEasy -1.2 This is a straightforward C2 integration question requiring only direct application of standard power rule integration to three simple terms, followed by substitution of limits. No problem-solving or insight needed—purely mechanical application of basic integration formulas.
Spec1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
Use calculus to find the exact value of \(\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
AnswerMarks Guidance
\(\int(3x^2 + 5 + 4x^{-2})dx = \frac{3x^3}{3} + 5x + \frac{4x^{-1}}{-1} = x^3 + 5x - 4x^{-1}\)M1, A1, A1 Accept any correct version, simplified or not. All 3 terms correct: M1 A1 A1. Two terms correct: M1 A1 A0. One power correct: M1 A0 A0. The given function must be integrated to score M1, not e.g. \(3x^4 + 5x^2 + 4\).
\([x^3 + 5x - 4x^{-1}]_1^2 = (8 + 10 - 2) - (1 + 5 - 4) = 14\)M1, A1 M1: Substituting 2 and 1 into a 'changed function' and subtracting, either way round.
(5 marks) 
| $\int(3x^2 + 5 + 4x^{-2})dx = \frac{3x^3}{3} + 5x + \frac{4x^{-1}}{-1} = x^3 + 5x - 4x^{-1}$ | M1, A1, A1 | Accept any correct version, simplified or not. All 3 terms correct: M1 A1 A1. Two terms correct: M1 A1 A0. One power correct: M1 A0 A0. The given function must be integrated to score M1, not e.g. $3x^4 + 5x^2 + 4$. |
| --- | --- | --- |
| $[x^3 + 5x - 4x^{-1}]_1^2 = (8 + 10 - 2) - (1 + 5 - 4) = 14$ | M1, A1 | M1: Substituting 2 and 1 into a 'changed function' and subtracting, either way round. |
| | (5 marks) | |
Use calculus to find the exact value of $\int _ { 1 } ^ { 2 } \left( 3 x ^ { 2 } + 5 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$.\\

\hfill \mbox{\textit{Edexcel C2 2006 Q2 [5]}}
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Q1 4 Q2 5 Q3 4 Q4 8 Q5 8 Q6 4 Q7 8 Q8 9 Q9 11 Q10 14