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A-LevelMathematicsIntegration9709/22 · May/June 2019 · Q48 Marks

Find the exact value of ∫(from 0 to π/2) (4 sin 2x + 2 cos²x)dx. Show all necessary working.

  1. (a)Find the exact value of ∫(from 0 to π/2) (4 sin 2x + 2 cos²x)dx. Show all necessary working.[5]
  2. (b)Use the trapezium rule with two intervals to find an approximation to ∫(from 2 to 8) √(ln(1+x)) dx.[3]

Mark scheme

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

  1. [B1] Use identity 2cos²x = 1 + cos2x
  2. [B1] Integrate to obtain form x + ½sin 2x
  3. [B1] Integrate to obtain -2cos2x
  4. [M1] Apply limits correctly, retaining exactness
  5. [A1] Obtain 4 + ½π or similarly simplified equivalent
  6. [B1] Use y values √ln3, √ln6, √ln9 or decimal equivalents
  7. [M1] Use correct formula, or equivalent, with h = 3, and three y values
  8. [A1] Obtain ½ × 3(√ln3+ 2√ln6 + √ln9) and hence 7.81

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, May/June 2019 Paper 2 Variant 2, question 4.

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