CAIE P3 2007 November — Question 3 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2007
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Parts
TypeShow that integral equals expression
DifficultyStandard +0.3 This is a straightforward application of integration by parts with ln x, a standard technique taught in P3. The question requires choosing u = ln x and dv = dx, then evaluating the definite integral with simple arithmetic involving logarithms. While it requires careful execution, it's a routine textbook exercise with no problem-solving insight needed, making it slightly easier than average.
Spec1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives

3 Use integration by parts to show that $$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$

AnswerMarks Guidance
Using 1 and ln \(x\) as parts reach \(x\ln x - \int x \cdot \frac{1}{x}dx\)M1*
Obtain indefinite integral \(x\ln x - x\)A1
Substitute correct limits correctlyM1(dep*)
Obtain given answerA1 [4]
Using 1 and ln $x$ as parts reach $x\ln x - \int x \cdot \frac{1}{x}dx$ | M1* |
Obtain indefinite integral $x\ln x - x$ | A1 |
Substitute correct limits correctly | M1(dep*) |
Obtain given answer | A1 | [4]
3 Use integration by parts to show that

$$\int _ { 2 } ^ { 4 } \ln x \mathrm {~d} x = 6 \ln 2 - 2$$

\hfill \mbox{\textit{CAIE P3 2007 Q3 [4]}}