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0606/22 Oct/Nov 2025, Question 7: Integrating eˣ, Trig and (ax+b)ⁿ

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 6 marks

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The question, in short

Given y = x cos 2x, differentiate it, then use that result to work out the integral of x sin 2x.

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

Part (a) · 2 marks

Find dy/dx for y = x cos 2x.

  1. 1Product rule with u = x, v = cos 2x
  2. 2d/dx(cos 2x) = −2 sin 2x
  3. 3dy/dx = cos 2x − 2x sin 2x

Answerdy/dx = cos 2x − 2x sin 2x

Part (b) · 4 marks

Using part a, find ∫ x sin 2x dx.

  1. 1From (a): ∫ (cos 2x − 2x sin 2x) dx = x cos 2x
  2. 2So −2∫ x sin 2x dx = x cos 2x − ∫ cos 2x dx
  3. 3−2∫ x sin 2x dx = x cos 2x − (1/2) sin 2x
  4. 4∫ x sin 2x dx = −(1/2)x cos 2x + (1/4) sin 2x + c

Answer−(1/2)x cos 2x + (1/4) sin 2x + c

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Also searched as: 0606/22/O/N/25 · 0606_w25_qp_22 · 0606 w25 22 · question 7.
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