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A-LevelMathematicsIntegration9709/31 · May/June 2020 · Q78 Marks

Let f(x) = cos x / (1 + sin x)

  1. (a)Show that f'(x) < 0 for all x in the interval −½π < x < ³⁄₂π.[4]
  2. (b)Find ∫ from π/6 to π/2 of f(x) dx. Give your answer in a simplified exact form.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

  1. [M1] (a) Use quotient or product rule
  2. [A1] (a) Obtain derivative in any correct form e.g. (-sin x (1+sin x) - cos x (cos x)) / (1+sin x)²
  3. [M1] (a) Use Pythagoras to simplify the derivative
  4. [A1] (a) Justify the given statement
  5. [*M1] (b) State integral of the form a ln (1 + sin x)
  6. [A1] (b) State correct integral ln (1 + sin x)
  7. [DM1] (b) Use limits correctly
  8. [A1] (b) Obtain answer ln (4/3)

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, May/June 2020 Paper 3 Variant 1, question 7.

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