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7 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2 · Question 2(a)(ii)

Let F_n(x) = \int (\ln x)^n \, dx.

Hence, or otherwise, show that F_3(2) - F_3(1) = 2(\ln 2)^3 - 6(\ln 2)^2 + 12\ln 2 - 6.

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  1. 2(a)(i)Show that F_n(x) = x(\ln x)^n - n F_{n-1}(x).[3 marks]
  2. 2(b)(i)By decomposing \frac{y^2 + 2y + 1}{y^4 + 2y^2 + 1} into partial fractions, show that…[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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