CAIE A-Level Mathematics 3.5.4 An integrand of the form x

CAIE A-Level Mathematics 3.5.4 An integrand of the form x
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise recognising f'(x)/f(x) structures in trig and rational integrands, then integrating to logarithmic exact values.

How this is tested

  • recognise f'(x)/f(x) patterns before integrating to a logarithmic expression
  • use exact trigonometric limits to simplify definite integral answers

Question 6(b)

[Maximum number: 4]

Hence show that ∫14π13π(cosec⁡2θ−cot⁡2θ)dθ=12ln⁡2\int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi}(\operatorname{cosec} 2 \theta-\cot 2 \theta) \mathrm{d} \theta=\frac{1}{2} \ln 2.