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A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2017 · Q811 Marks

Let Iⁿ = ∫(from 0 to π/4) secⁿx dx for n > 0.

  1. (i)Find the value of I₂.[2]
  2. (ii)Show that, for n > 2, (n - 1)Iⁿ = 2ⁿ⁻¹ + (n - 2)Iⁿ⁻².[5]
  3. (iii)The curve C has equation y = sec³x for 0 ≤ x ≤ π/4. The region R is bounded by C, the x-axis, the y-axis and the line x = π/4. Find the volume of revolution generated when R is rotated through 2π radians about the x-axis.[4]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

  1. [2] M1A1
  2. [1] M1 For setting up integration by parts (u = secⁿ⁻²x, dv/dx = sec²x)
  3. [2] M1A1 For applying integration by parts: [secⁿ⁻²x tanx] - ∫ (n-2)secⁿ⁻²x tan²x dx
  4. [1] M1 For using tan²x = sec²x - 1 and splitting the integral
  5. [1] A1 AG For rearranging and showing (n - 1)Iⁿ = 2ⁿ⁻¹ + (n - 2)Iⁿ⁻²
  6. [1] M1
  7. [1] M1
  8. [1] M1
  9. [1] A1

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About this question

Topic
Integration, A-Level Further Mathematics (syllabus 9231). Structured question, 11 marks.
Source
A-Level Further Mathematics, Oct/Nov 2017 Paper 1 Variant 2, question 8.

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