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5 marksIntegration II

CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2 · Question 2(b)

The gradient at the point (x, y) on a curve is given by \frac{\mathrm{d}y}{\mathrm{d}x} = e^{4x}. Given that the curve passes through the point (0, 1), find its equation.

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Other parts of this question

  1. 2(a)Solve the differential equation \frac{\mathrm{d}y}{\mathrm{d}x} + y = e^{2x}.[5 marks]
  2. 2(c)Evaluate \int_1^e x^2 \ln x \,\mathrm{d}x, writing your answer in terms of e.[7 marks]
  3. 2(d)(i)Use the substitution v = 1 - u to find \int \frac{\mathrm{d}u}{\sqrt{1 - u}}.[3 marks]
  4. 2(d)(ii)Hence, or otherwise, use the substitution u = \sin x to evaluate \int_0^{\pi/2} \sqrt{1 + \sin x} \,\mathrm{d}x.[5 marks]

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