A-LevelMathematicsIntegration9709/23 · May/June 2017 · Q79 Marks

Find ∫ (2 cos θ – 3)(cos θ + 1) dθ.

(a)Find ∫ (2 cos θ – 3)(cos θ + 1) dθ.[4]
(b)(i)Find ∫ (4/(2x+1) + 1/(2x)) dx.[2]
(b)(ii)Hence find ∫ from 1 to 4 of (4/(2x+1) + 1/(2x)) dx, giving your answer in the form ln k.[3]

Mark scheme

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

[B1] Obtain ∫(2cos²θ-cosθ-3)dθ
[M1] Attempt use of identity to obtain integrand involving cos 2θ and cosθ
[M1] Integrate to obtain form k₁ sin 2θ+ k₂ sin θ+ k₃θ for non-zero constants
[A1] Obtain 1/2 sin 2θ – sin θ – 2θ + c
[M1] Integrate to obtain form k₁ ln(2x+1)+ k₂ ln(x) or k₁ ln(2x+1)+k₂ ln(2x)
[A1] Obtain 2ln(2x+1)+1/2lnx or 2ln(2x+1)+1/2ln(2x)
[M1] Use relevant logarithm power law for expression obtained from application of limits
[M1] Use relevant logarithm addition / subtraction laws
[A1] Obtain ln18

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source: A-Level Mathematics, May/June 2017 Paper 2 Variant 3, question 7.

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