2024 IGCSE Additional Mathematics May/June 0606/11 Q12

2024 IGCSE Additional Mathematics May/June 0606/11 Q12

8 marks
  1. It is given that \(y=\dfrac{\ln3x}{x^2}\) for \(x>0\).
    Find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\). Give your answer in the form \(\dfrac{A+B\ln3x}{x^3}\), where \(A\) and \(B\) are integers.[4]
  2. Hence find \(\displaystyle\int\dfrac{\ln3x}{x^3}\,\mathrm{d}x\).[4]

Solution:

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Answer:(a) \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{1-2\ln3x}{x^3}\), \(A=1,\ B=-2\). (b) \(-\dfrac1{4x^2}-\dfrac{\ln3x}{2x^2}+C\).

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