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6 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2026 · Paper 2 · Question 2(c)(i)

It is given that I_n = ∫(from 0 to π/2) sin^n θ dθ for integers n ≥ 0.

Show that I_n = (n-1)/n * I_(n-2).

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Other parts of this question

  1. 2(a)(i)Apply the trapezium rule with THREE equal intervals to estimate the value of ∫(from 0 to π/3) tan x dx.[4 marks]
  2. 2(a)(ii)Using an appropriate trigonometric identity, determine the EXACT value of ∫(from 0 to π/3) tan x dx.[4 marks]
  3. 2(b)Use the substitution u = cos^(-1)(x/2) to show that ∫(from 0 to 1) (cos^(-1)(x/2))/√(4-x^2) dx = 5π^2/72.[7 marks]
  4. 2(c)(ii)a)Verify that I_0 = π/2.[2 marks]
  5. 2(c)(ii)b)Hence, determine the value of I_4.[2 marks]

More practice: the rest of this paper · more Integration II questions · all CAPE Pure Mathematics Unit 2 past papers