CAIE A-Level Mathematics 3.5.3 Integration by Partial Fractions

CAIE A-Level Mathematics 3.5.3 Integration by Partial Fractions
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise decomposing rational functions and integrating linear factors to logarithms or irreducible quadratic factors to inverse tangents before applying exact limits.

How this is tested

  • decompose the rational function fully before integrating each partial-fraction term
  • integrate linear denominators as logarithms and quadratic forms as inverse tangents
  • apply transformed or original bounds and combine logarithms into the required exact form

Question 9(b)

[Maximum number: 4]

Let f(x)=x2+4ax+6a2(x+2a)(x+3a)\mathrm{f}(x)=\frac{x^{2}+4 a x+6 a^{2}}{(x+2 a)(x+3 a)}, where a is a positive constant.

Hence find the exact value of ∫−aaf(x)dx\int_{-a}^{a} \mathrm{f}(x) \mathrm{d} x. Give your answer in the form a(p+ln⁡q)a(p+\ln q), where p and q are rational.

Figure for Question 9(b) — CAIE A-Level Mathematics