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6 marksIntegration I

CAPE Pure Mathematics Unit 1 · May/June 2016 · Paper 2 · Question 5(c)

Given that ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a - x) dx for a > 0.

Show that ∫₀¹ (eˣ / (eˣ + e¹⁻ˣ)) dx = 1/2.

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Other parts of this question

  1. 5(a)Use an appropriate substitution to find ∫x(x + 1)³ dx.[4 marks]
  2. 5(b)Calculate the volume of the solid that results from rotating R about the y-axis.[5 marks]
  3. 5(d)(i)Solve an appropriate differential equation to show that the number of bacteria present at any time can be modelled by the equation y = 10,000 e⁰.⁰²ᵗ.[7 marks]
  4. 5(d)(ii)Determine the time required for the bacteria population to double in size.[3 marks]

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