Pre-U Pre-U 9794/2 2012 June — Question 3 4 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2012
SessionJune
Marks4
TopicIndefinite & Definite Integrals
TypeExponential and logarithmic integration
DifficultyEasy -1.2 This is a straightforward integration question requiring only basic techniques: integrating e^x (which gives e^x) and x (which gives x²/2), then evaluating at the limits 0 and 1. It's a routine 4-mark question with no problem-solving element, making it easier than average but not trivial since it requires careful handling of the exact value (e - 1/2).
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]
AnswerMarks Guidance
\(\int_0^1 (e^x - x)dx = \left[e^x - \frac{1}{2}x^2\right]_0^1\)M1, A1, M1 \(ke^x + mx^2\); Use of limits; Without \(+ c\)
\(= e - \frac{1}{2}\)A1 [4] [4] 
$\int_0^1 (e^x - x)dx = \left[e^x - \frac{1}{2}x^2\right]_0^1$ | M1, A1, M1 | $ke^x + mx^2$; Use of limits; Without $+ c$

$= e - \frac{1}{2}$ | A1 [4] | [4]
Find the exact value of $\int_0^1 (e^x - x) dx$. [4]

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q3 [4]}}
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