2026 NORTHBROOKS P1 Q3

2026 NORTHBROOKS P1 Q3

Secondary 4
8 marks
  1. Express \(\dfrac{7x^2-30x+3}{(x-5)(x^2+3)}\) in partial fractions.[5]
  2. Differentiate \(\ln(x^2+3)\) with respect to \(x\).[1]
  3. Use your answers to part (a) and (b) to find \(\displaystyle\int\dfrac{7x^2-30x+3}{(x-5)(x^2+3)}\,\mathrm{d}x\).[2]

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Answer:(a) \(\dfrac1{x-5}+\dfrac{6x}{x^2+3}\). (b) \(\dfrac{2x}{x^2+3}\). (c) \(\ln|x-5|+3\ln(x^2+3)+C\).

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