A-LevelMathematicsIntegration9709/22 · Oct/Nov 2019 · Q69 Marks

Show that ∫(from 2 to 18) (3/(2x)) dx = ln 27.

(a)Show that ∫(from 2 to 18) (3/(2x)) dx = ln 27.[4]
(b)Find the exact value of ∫(from 0 to π) 4 sin²(¾x) dx. Show all necessary working.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] Obtain (3/2)ln x or (3/2)ln(2x) or (3/2)ln(kx)
[M1] Use subtraction law of logarithms correctly, showing sufficient detail (e.g. ln 216 - ln 8 = ln(216/8))
[M1] Use power law of logarithms correctly (e.g. n ln(kx) = ln((kx)^n))
[A1] Confirm ln 27 with sufficient working and no incorrect working (AG)
[*M1] Use appropriate identity to express integrand in form k₁ + k₂ cos(3x) (k₁ ≠ 0. Allow 2 * (3/2)x for 3x)
[A1] Obtain correct 2 - 2cos(3x)
[DM1] Integrate to obtain form k₃x + k₄ sin(3x)
[A1] Obtain correct 2x - (2/3)sin(3x)
[A1] Use limits to obtain 2π + 4/3 or exact equivalent

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, Oct/Nov 2019 Paper 2 Variant 2, question 6.

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