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5 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2 · Question 2(b)

Let f(x) = x|x| = \begin{cases} x^2 &; x \ge 0 \\ -x^2 &; x < 0 \end{cases}. Use the trapezium rule with four intervals to calculate the area between f(x) and the x-axis for the domain -0.75 \le x \le 2.25.

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Other parts of this question

  1. 2(a)(i)Determine \int \sin x \cos 2x \, dx.[5 marks]
  2. 2(a)(ii)Hence, calculate \int_0^{\frac{\pi}{2}} \sin x \cos 2x \, dx.[2 marks]
  3. 2(c)(i)Show that \frac{2x^2 + 4}{(x^2 + 4)^2} = \frac{2}{x^2 + 4} - \frac{4}{(x^2 + 4)^2}.[6 marks]
  4. 2(c)(ii)Hence, find \int \frac{2x^2 + 4}{(x^2 + 4)^2} \, dx. Use the substitution x = 2\tan\theta.[7 marks]

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