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7 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2 · Question 2(b)(i)

By decomposing \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} into partial fractions, show that \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} = \frac{1}{y^2 + 1} + \frac{2y}{(y^2 + 1)^2}.

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Other parts of this question

  1. 2(a)(i)Show that F_n(x) = x(\ln x)^n - n F_{n-1}(x).[3 marks]
  2. 2(a)(ii)Hence, or otherwise, show that F_3(2) - F_3(1) = 2(\ln 2)^3 - 6(\ln 2)^2 + 12\ln 2 - 6.[7 marks]
  3. 2(b)(ii)Hence, find \int_0^1 \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} \, dy.[8 marks]

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