O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2017 · Q98 Marks

Find d/dx (x ln x).

(i)Find d/dx (x ln x).[2]
(ii)Hence find ∫ ln x dx.[2]
(iii)Hence, given that k > 0, show that ∫ (from k to 2k) ln x dx = k(ln 4k - 1).[4]

Mark scheme

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

[M1A1] d/dx (xlnx) = x(1/x) + lnx isw; Product rule. One correct term + another term. Allow unsimplified.
[M1] ∫lnxdx = xlnx - x + (C); Correct use of (i) and must be dealing with 2 terms. soi
[A1] Correct answer with no working is fine.
[M1] ∫(from k to 2k) lnx dx = [2kln2k - 2k] - [klnk -k]; Insert limits and subtract correctly using their result from (ii) which must contain an ln function
[M1] = k(2ln2k – lnk – 1); Uses nlna = lna^n somewhere oe
[M1] = k(ln(2k)² - Ink-1); Uses lna - lnb = ln(a/b) or lna + lnb = lnab somewhere
[A1] = k(ln4k-1); Answer given Correct completion.

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Topic: Differentiation and integration, O-Level Additional Mathematics (syllabus 4037). Structured question, 8 marks.
Source: O-Level Additional Mathematics, Oct/Nov 2017 Paper 2 Variant 2, question 9.

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