O-LevelAdditional MathematicsDifferentiation and integration4037/12 · Oct/Nov 2010 · Q1110 Marks
Find ∫ 1/(√1 + x) dx.
- (i)Find ∫ 1/(√1 + x) dx.[2]
- (ii)Given that y = 2x/(√1 + x), show that dy/dx = A/(√1 + x) + Bx/(√1 + x)³, where A and B are to be found.[4]
- (iii)Hence find ∫ 2x/(√1 + x)³ dx and evaluate ∫ (from 0 to 3) x/(√1 + x)³ dx.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2010). The mark or mark type is shown in brackets.
- [M1, A1] (i) 2(1+x)^(1/2) (+ c) (M1 for (1+x)^(1/2), A1 for 2)
- [M1] (ii) dy/dx = (2√1+x - 2x(1/2)(1+x)^(-1/2)) / (1+x) (M1 attempt at differentiation)
- [A2, 1, 0] (ii) A1 all correct (for A and B, -1 each error in derivation)
- [M1] (iii) M1 for idea of using (ii) 'in reverse'
- [A1] (iii) ∫ x/(√1+x)³ dx = 4√1+x - 2x/(√1+x) (+c) (A1 all correct)
- [M1, A1] (iii) ∫ (from 0 to 3) x/(√1 + x)³ dx = (8-3) - (4) = 1 (M1 for attempt evaluation)
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