A-LevelMathematicsIntegration9709/22 · 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 ∫¹⁴ (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 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 2, question 7.

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