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0606/23 May/Jun 2026, Question 10: Integrating eˣ, Trig and (ax+b)ⁿ

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 6 marks

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The question, in short

Differentiate x sin(2x − 3), then use the result to evaluate the definite integral of x cos(2x − 3) from 0 to 1/2.

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

Part (a) · 2 marks

Find the derivative of x sin(2x − 3) with respect to x.

  1. 1Product rule: d/dx[x] × sin(2x − 3) + x × d/dx[sin(2x − 3)]
  2. 2= sin(2x − 3) + 2x cos(2x − 3)

Answersin(2x − 3) + 2x cos(2x − 3)

Part (b) · 4 marks

Using part a, evaluate the integral of x cos(2x − 3) between limits 0 and 1/2.

  1. 1From (a): 2x cos(2x − 3) = d/dx[x sin(2x − 3)] − sin(2x − 3)
  2. 2∫ x cos(2x − 3) dx = (1/2)[x sin(2x − 3) + (1/2)cos(2x − 3)]
  3. 3At x = 1/2: (1/2)[(1/2)sin(−2) + (1/2)cos(−2)]; at x = 0: (1/2)[(1/2)cos(−3)]
  4. 4= (1/4)(−sin 2 + cos 2 − cos 3)
  5. 5= (1/4)(−0.9093 − 0.4161 + 0.9900) = −0.0839

Answer−0.0839 (exact: (cos 2 − sin 2 − cos 3)/4)

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Also searched as: 0606/23/M/J/26 · 0606_s26_qp_23 · 0606 s26 23 · question 10.
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