CAIE A-Level Mathematics 3.5.2 Trigonometric Integration

CAIE A-Level Mathematics 3.5.2 Trigonometric Integration
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise rewriting powers, products and quotients of sine, cosine, tangent and secant with identities before integrating and evaluating exact trigonometric bounds.

How this is tested

  • use double-angle or power-reduction identities to rewrite even powers of sine or cosine
  • convert tan² θ to sec² θ − 1 or simplify a quotient before integrating
  • substitute radian bounds into the transformed antiderivative and simplify exact π or surd terms

Question 7(b)

[Maximum number: 6]

Hence find the exact value of ∫−18π18π(cos⁡4θ−sin⁡4θ+4sin⁡2θcos⁡2θ)dθ\int_{-\frac{1}{8} \pi}^{\frac{1}{8} \pi}\left(\cos ^{4} \theta-\sin ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta\right) \mathrm{d} \theta.