O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2021 · Q108 Marks
Find ∫(e³x+1)³dx.
- (a)Find ∫(e³x+1)³dx.[2]
- (b)(i)Differentiate, with respect to x, y = x sin 4x.[2]
- (b)(ii)Hence show that ∫(from π/4 to π/3) 4x cos4xdx = 1/8 - (π√3)/6.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.
- (a) B2 B1 for ke³x+3 or ke³xe³x where k ≠ 1/3 or 0
- (a) 1/3 e³x+3 + c or 1/3 e³xe³x + c nfww
- (b)(i) B1 d(sin4x)/dx = 4 cos4x soi
- (b)(i) B1 FT their 4 cos 4x if possible Applies correct form of product rule: 4x cos 4x + [1] sin 4x isw
- (b)(ii) M1 FT use of their mx cos 4x + n sin 4x where m and n are constants ∫(4x cos4x)dx = xsin 4x - ∫sin 4xdx
- (b)(ii) A1 x sin 4x + 1/4 cos 4x [+c] soi
- (b)(ii) A1 [ (π/3)sin(4π/3) + (1/4)cos(4π/3) ] - [ (π/4)sin(4π/4) + (1/4)cos(4π/4) ]
- (b)(ii) A1 Correct completion to given answer 1/8 - (π√3)/6
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