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5 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2022 · Paper 2 · Question 2(b)(i)

Show that if I_n = ∫ (1 - x)ⁿeᵃˣ dx for n ≥ 1, then I_n = (1 - x)ⁿeᵃˣ/a + n/a I_n-1.

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Other parts of this question

  1. 2(a)(i)Use partial fractions to show that x⁴ / (x⁴ - 1) = 1 + 1/(4(x-1)) - 1/(4(x+1)) - 1/(2(x²+1)).[9 marks]
  2. 2(a)(ii)Hence, or otherwise, determine ∫ x⁴ / (x⁴ - 1) dx.[6 marks]
  3. 2(b)(ii)Hence, determine ∫ (1 - x)²eᵃˣ dx.[5 marks]

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