O-LevelAdditional MathematicsDifferentiation and integration4037/11 · May/June 2016 · Q58 Marks
Do not use a calculator in this question.
- (i)Show that d/dx (e^(4x)/4 - x e^(4x)) = pxe^(4x), where p is an integer to be found.[4]
- (ii)Hence find the exact value of ∫(from 0 to ln2) xe^(4x) dx, giving your answer in the form a ln2 + b/c, where a, b and c are integers to be found.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [1] d/dx (e^(4x)/4) = e^(4x)
- [1] d/dx (x e^(4x)) = (x * 4e^(4x)) + e^(4x)
- [1] e^(4x) - ((x * 4e^(4x)) + e^(4x))
- [1] = -4xe^(4x), so p = -4
- [1] Integral from 0 to ln2 of xe^(4x) dx = -1/4 [e^(4x)/4 - xe^(4x)](from 0 to ln2)
- [1] e^(4ln2) = 16
- [1] = -1/4 * [ (16/4 - 2ln2*16) - (1/4 - 0) ]
- [1] = 4ln2 - 15/16
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