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5 marksIntegration I

CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 2 · Question 6(a)(i)

By using the substitution u = 1 − x, find ∫ x (1-x)² dx.

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  1. 6(a)(ii)show that ∫[f(t) + g(t)] dt = ∫f(t) dt + ∫g(t) dt.[4 marks]
  2. 6(b)(i)Show that r = (600-2x) / (2 + π)[2 marks]
  3. 6(b)(ii)Hence, determine the length, x, that maximises the area enclosed by the track.[6 marks]
  4. 6(c)(i)Show that y'' = x sin x.[4 marks]
  5. 6(c)(ii)Hence, determine the specific solution of the differential equation y'' = x sin x, given that when x = 0, y = 1 and when x = π, y = 6.[4 marks]

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