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3 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 2 · Question 2(a)(i)

Let F_n(x) = ∫(ln x)ⁿ dx.

Show that F_n(x) = x (ln x)ⁿ - nF_(n-1)(x).

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Other parts of this question

  1. 2(a)(ii)Hence, or otherwise, show that F₂(2) - F₂(1) = 2 (ln 2)³ - 6 (ln 2)² + 12 ln 2 - 6.[7 marks]
  2. 2(b)(i)By expressing (y² + 2y + 1) / (y⁴ + 2y² + 1) as partial fractions, show that (y² + 2y + 1) / (y⁴ + 2y² + 1) = 1 / (y² + 1) + 2y / (y² + 1)².[7 marks]
  3. 2(b)(ii)Hence, or otherwise, evaluate ∫(y² + 2y + 1) / (y⁴ + 2y² + 1) dy.[8 marks]

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