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5 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2 · Question 2(c)(i)

Use partial fractions to show that \frac{2x^2 - x + 4}{x^3 + 4x} = \frac{1}{x} + \frac{x}{x^2 + 4} - \frac{1}{x^2 + 4}.

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  1. 2(a)(i)Use integration by parts to derive the reduction formula aI_n = x^n e^{ax} - nI_{n-1}, where I_n = \int x^n e^{ax} \, \mathrm{d}x.[4 marks]
  2. 2(a)(ii)Hence, or otherwise, determine \int x^3 e^{3x} \, \mathrm{d}x.[6 marks]
  3. 2(b)Calculate \int_0^1 \frac{\sin^{-1} x}{\sqrt{1 - x^2}} \, \mathrm{d}x.[5 marks]
  4. 2(c)(ii)Hence, or otherwise, determine \int \frac{2x^2 - x + 4}{x^3 + 4x} \, \mathrm{d}x.[5 marks]

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