Question 1 - A-Level Maths

CAIE P2 2007 November — Question 1 4 marks

Exam Board	CAIE
Module	P2 (Pure Mathematics 2)
Year	2007
Session	November
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Standard Integrals and Reverse Chain Rule
Type	Definite integral with logarithmic form
Difficulty	Moderate -0.8 This is a straightforward application of the standard integral ∫1/(ax+b)dx = (1/a)ln\|ax+b\| + c, requiring only recognition of the form and careful substitution of limits. The arithmetic simplifies nicely (ln9 - ln3 = ln3), making this easier than average with minimal problem-solving required.
Spec	1.06f Laws of logarithms: addition, subtraction, power rules 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits

1 Show that $$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3$$

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Answer	Marks	Guidance
State indefinite integral of the form $k \ln(2x + 1)$, where $k = \frac{1}{2}, 1$ or $2$	M1
State correct integral $\frac{1}{2}\ln(2x + 1)$	A1
Use limits correctly, allow use of limits $x = 4$ and $x = 1$ in an incorrect form	M1
Obtain given answer	A1	[4]

State indefinite integral of the form $k \ln(2x + 1)$, where $k = \frac{1}{2}, 1$ or $2$ | M1 |
State correct integral $\frac{1}{2}\ln(2x + 1)$ | A1 |
Use limits correctly, allow use of limits $x = 4$ and $x = 1$ in an incorrect form | M1 |
Obtain given answer | A1 | [4]

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1 Show that

$$\int _ { 1 } ^ { 4 } \frac { 1 } { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 2 } \ln 3$$

\hfill \mbox{\textit{CAIE P2 2007 Q1 [4]}}

This paper (8 questions)

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Q1 4 Q2 5 Q3 5 Q4 5 Q5 6 Q6 7 Q7 8 Q8 10

Answer	Marks	Guidance
State indefinite integral of the form \(k \ln(2x + 1)\), where \(k = \frac{1}{2}, 1\) or \(2\)	M1
State correct integral \(\frac{1}{2}\ln(2x + 1)\)	A1
Use limits correctly, allow use of limits \(x = 4\) and \(x = 1\) in an incorrect form	M1
Obtain given answer	A1	[4]