O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2017 · Q55 Marks

Show that d/dx [0.4x⁵ (0.2 - ln 5x)] = kx⁴ ln 5x, where k is an integer to be found.

(i)Show that d/dx [0.4x⁵ (0.2 - ln 5x)] = kx⁴ ln 5x, where k is an integer to be found.[2]
(ii)Express ln 125x³ in terms of ln 5x.[1]
(iii)Hence find ∫ (x⁴ ln 125x³)dx.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.

[M1] (their 0.4x⁴)(0.2 - ln 5x) + 0.4x⁵ (their -5/(5x)) oe or their 0.4x⁴ - (their 2x⁴)ln 5x + 0.4x⁵ (their -5/(5x)) oe (clearly applies correct form of product rule)
[A1] -2x⁴ ln 5x isw
[B1] 3ln 5x or ln 5x + ln 5x + ln 5x
[M1] FT k = 2 from (i) allow for -3/2 ∫ (2x⁴ ln 5x)dx or, when k = -2, for ∫ (x⁴ ln 5x)dx = -0.2x⁵(0.2 - ln 5x) or -2/3 ∫ (3x⁴ ln 5x)dx = 0.4x⁵(0.2 - ln 5x) oe or, when FT k = 2, for ∫ (x⁴ ln 5x)dx = 0.2x⁵(0.2 - ln 5x) or 2/3 ∫ (3x⁴ ln 5x)dx = 0.4x⁵(0.2 - ln 5x) oe
[A1] -3/2 (0.4x⁵(0.2-ln5x)) [+] oe isw cao nfww; implies M1 An answer of 0.6x⁵(0.2-ln5x) following k = 2 from (i) implies M1 A0

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 5 marks.
Source: O-Level Additional Mathematics, May/June 2017 Paper 2 Variant 2, question 5.

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