CAIE P2 2010 June — Question 2 4 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2010
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeDefinite integral with logarithmic form
DifficultyModerate -0.8 This is a straightforward application of the standard integral ∫(1/x)dx = ln|x| + c with a linear substitution. Students need only recognize the form, integrate to get ln|x+2|, apply limits 0 and 6, and simplify ln(8) - ln(2) = ln(4) = 2ln(2). It's a routine textbook exercise testing basic logarithmic integration with minimal algebraic manipulation, making it easier than average.
Spec1.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)

2 Show that \(\int _ { 0 } ^ { 6 } \frac { 1 } { x + 2 } \mathrm {~d} x = 2 \ln 2\).

AnswerMarks Guidance
Obtain integral \(\ln(x + 2)\)B1
Substitute correct limits correctlyM1
Use law for the logarithm of a product, quotient or a powerM1
Obtain given answer following full and correct workingA1 [4]
Obtain integral $\ln(x + 2)$ | B1 |
Substitute correct limits correctly | M1 |
Use law for the logarithm of a product, quotient or a power | M1 |
Obtain given answer following full and correct working | A1 | [4]
2 Show that $\int _ { 0 } ^ { 6 } \frac { 1 } { x + 2 } \mathrm {~d} x = 2 \ln 2$.

\hfill \mbox{\textit{CAIE P2 2010 Q2 [4]}}