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4 marksDifferentiation II

CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2 · Question 2(a)(i)

Given that n is a positive integer, find \frac{\mathrm{d}}{\mathrm{d}x}[x(\ln x)^n].

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  1. 2(a)(ii)Hence, or otherwise, derive the reduction formula I_n = x(\ln x)^n - n I_{n-1}, where I_n = \int (\ln x)^n\,\mathrm{d}x.[4 marks]
  2. 2(a)(iii)Use the reduction formula in (a)(ii) to determine \int (\ln x)^3\,\mathrm{d}x.[6 marks]
  3. 2(b)(i)Using a suitable integrating factor, show that the general solution of this differential equation is y = t + 10 + \frac{c}{(t+10)^2}, where c is an…[7 marks]
  4. 2(b)(ii)Given that the tank initially contains 5\text{ kg} of salt in the liquid, calculate the amount of salt that dissolves in the tank of water at t = 15.[4 marks]

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