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A-LevelMathematicsIntegration9709/33 · May/June 2020 · Q79 Marks

Let f(x) = 2 / ((2x−1)²(2x + 1)).

  1. (a)Express f(x) in partial fractions.[2]
  2. (b)Using your answer to part (a), show that (f(x))² = 1/((2x−1)²) - 1/(2x−1) + 1/(2x+1) + 1/((2x + 1)²).[2]
  3. (c)Hence show that ∫₁² (f(x))² dx = ⁸/₁₅ + ½ ln(⁵/₃).[5]

Mark scheme

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

  1. [M1] State or imply the form A/(2x-1) + B/(2x+1) and use a relevant method to find A or B
  2. [A1] Obtain A = 1, B = -1
  3. [M1] Square the result of part (a) and substitute the fractions of part (a)
  4. [A1] Obtain the given answer correctly
  5. [B3, 2, 1, 0] Integrate and obtain -1/(2(2x-1)) - ½ln(2x-1) + ½ln(2x+1) - 1/(2(2x+1)), or equivalent
  6. [M1] Substitute limits correctly
  7. [A1] Obtain the given answer correctly

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About this question

Topic
Integration, A-Level Mathematics (syllabus 9709). Structured question, 9 marks.
Source
A-Level Mathematics, May/June 2020 Paper 3 Variant 3, question 7.

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