CAIE P2 2010 November — Question 3 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2010
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeDefinite integral with exponentials
DifficultyModerate -0.8 This is a straightforward integration question requiring only algebraic expansion of (e^x + 1)^2 = e^(2x) + 2e^x + 1, followed by standard integration and substitution of limits. The techniques are routine for P2 level with no problem-solving insight needed, making it easier than average but not trivial due to the exponential terms and arithmetic involved.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits

3 Show that \(\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }\).

AnswerMarks Guidance
Integrate and obtain \(\frac{1}{2}e^{2x}\) termB1
Obtain \(2e^x\) termB1
Obtain \(x\)B1
Use limits correctly, allow use of limits \(x = 1\) and \(x = 0\) into an incorrect formM1
Obtain given answerA1 [5]
S. R. Feeding limits into original integrand, 0/5
Integrate and obtain $\frac{1}{2}e^{2x}$ term | B1 |

Obtain $2e^x$ term | B1 |

Obtain $x$ | B1 |

Use limits correctly, allow use of limits $x = 1$ and $x = 0$ into an incorrect form | M1 |

Obtain given answer | A1 | [5]

S. R. Feeding limits into original integrand, 0/5
3 Show that $\int _ { 0 } ^ { 1 } \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } + 2 \mathrm { e } - \frac { 3 } { 2 }$.

\hfill \mbox{\textit{CAIE P2 2010 Q3 [5]}}