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7 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 2(a)(ii)

Let F_n(x) = \frac{1}{n!} \int_0^x t^n e^{-t} dt.

Show that F_n(x) = F_{n-1}(x) - \frac{1}{n!} x^n e^{-x}.

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Other parts of this question

  1. 2(a)(i)Find F_0(x) and F_n(0), given that 0! = 1.[3 marks]
  2. 2(a)(iii)Hence, show that if M is an integer greater than 1, then e^x F_M(x) = -\left(x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^M}{M!}\right) + (e^x - 1).[4 marks]
  3. 2(b)(i)Express \frac{2x^2 + 3}{(x^2 + 1)^2} in partial fractions.[5 marks]
  4. 2(b)(ii)Hence, find \int \frac{2x^2 + 3}{(x^2 + 1)^2} dx.[6 marks]

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