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7 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2 · Question 2(c)(ii)

Let I_n = \int_0^{\frac{\pi}{4}} \tan^n x\,dx, n \ge 2.

Hence evaluate I_4.

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Other parts of this question

  1. 2(a)Show that for n \ge 2, \tan^n x = \tan^{n-2} x \sec^2 x - \tan^{n-2} x.[3 marks]
  2. 2(b)Find \frac{dy}{dx} when y = \tan^n x.[3 marks]
  3. 2(c)(i)By using the result in (a) above, show that I_n + I_{n-2} = \frac{1}{n-1}.[7 marks]

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