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A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2018 · Q1214 Marks

OR Let I_n = ∫(from 1 to √2) (x² - 1)ⁿ dx.

  1. (i)Show that, for n ≥ 1, (2n + 1)I_n = √2 – 2nI_n₋₁.[5]
  2. (ii)Using the substitution x = sec θ, show that I_n = ∫(from 0 to π/4) tan²ⁿ⁺¹ θ sec θ dθ.[4]
  3. (iii)Deduce the exact value of ∫(from 0 to π/4) sin⁷ θ / cos⁸ θ dθ.[5]

Mark scheme

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

  1. [M1A1] Integrates by parts.
  2. [M1] Uses x² = x² −1+1.
  3. [A1]
  4. [A1] AG.
  5. [M1A1] Differentiates secθ.
  6. [B1] Changes limits.
  7. [B1] Uses sec²θ−1 = tan² θ . (AG.)
  8. [B1] Deduces that integral is I₃.
  9. [B1] Calculates I₀ or I₁
  10. [M1A1] Uses reduction formula to find I₁ or I₂
  11. [A1] Finds I₂.
  12. [A1] Finds I₃.

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About this question

Topic
Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 14 marks.
Source
A-Level Further Mathematics, Oct/Nov 2018 Paper 1 Variant 2, question 12.

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