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A-LevelMathematicsIntegration9709/31 · May/June 2018 · Q58 Marks

Let I = ∫(from 1/4 to 3/4) √((x)/(1-x)) dx.

  1. (i)Using the substitution x = cos² θ, show that I = ∫(from π/6 to π/3) 2 cos² θ dθ.[4]
  2. (ii)Hence find the exact value of I.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

  1. [B1] State or imply dx = −2 cos θ sin θ dθ, or equivalent
  2. [M1] Substitute for x and dx, and use Pythagoras
  3. [A1] Obtain integrand ±2cos²θ
  4. [A1] Justify change of limits and obtain given answer correctly
  5. [M1*] Obtain indefinite integral of the form aθ + b sin 2θ
  6. [A1] Obtain θ + (1/2)sin 2θ
  7. [M1(dep*)] Use correct limits correctly
  8. [A1] Obtain answer π/6 with no errors seen

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source
A-Level Mathematics, May/June 2018 Paper 3 Variant 1, question 5.

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