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0606/12 Feb/Mar 2025, Question 12: Integrating eˣ, Trig and (ax+b)ⁿ

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 7 marks

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The question, in short

Given y = xe^(3x+2), differentiate it, then use that result to work out the integral of xe^(3x+2) with respect to x.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Find dy/dx for y = xe^(3x+2).

  1. 1d/dx(e^(3x+2)) = 3e^(3x+2)
  2. 2Product rule: dy/dx = x × 3e^(3x+2) + 1 × e^(3x+2)
  3. 3dy/dx = 3xe^(3x+2) + e^(3x+2)

Answerdy/dx = 3xe^(3x+2) + e^(3x+2)

Part (b) · 4 marks

Using your answer to part a, find ∫ xe^(3x+2) dx.

  1. 1From (a): ∫ (3xe^(3x+2) + e^(3x+2)) dx = xe^(3x+2)
  2. 2∫ 3xe^(3x+2) dx = xe^(3x+2) − ∫ e^(3x+2) dx
  3. 3∫ 3xe^(3x+2) dx = xe^(3x+2) − (1/3)e^(3x+2)
  4. 4∫ xe^(3x+2) dx = (1/3)xe^(3x+2) − (1/9)e^(3x+2) + c

Answer(1/3)xe^(3x+2) − (1/9)e^(3x+2) + c

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