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0606/23 Oct/Nov 2025, Question 4: Integrating eˣ, Trig and (ax+b)ⁿ

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 7 marks

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The question, in short

Integrate two expressions in x, then show a definite integral of sec²(x/2) from π/3 to π/2 equals 2(1 − √3/3).

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

Part (ai) · 2 marks

Integrate e^(5x−2) with respect to x.

  1. 1∫ e^(ax+b) dx = (1/a)e^(ax+b) + c
  2. 2∫ e^(5x−2) dx = (1/5)e^(5x−2) + c

Answer(1/5)e^(5x−2) + c

Part (aii) · 2 marks

Integrate 1/(4 − 3x) with respect to x, given x < 4/3.

  1. 1∫ 1/(ax + b) dx = (1/a) ln(ax + b) + c, with a = −3
  2. 2∫ 1/(4 − 3x) dx = −(1/3) ln(4 − 3x) + c

Answer−(1/3) ln(4 − 3x) + c

Part (b) · 3 marks

Show that the integral of sec²(x/2) from π/3 to π/2 equals 2(1 − √3/3).

  1. 1∫ sec²((1/2)x) dx = 2 tan((1/2)x)
  2. 2[2 tan((1/2)x)] from π/3 to π/2 = 2 tan(π/4) − 2 tan(π/6)
  3. 3= 2(1 − 1/√3) = 2(1 − √3/3)

Answer∫ from π/3 to π/2 of sec²((1/2)x) dx = 2(1 − √3/3) shown

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Also searched as: 0606/23/O/N/25 · 0606_w25_qp_23 · 0606 w25 23 · question 4.
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