CAIE A-Level Mathematics 3.5 Integration Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise choosing substitution, integration by parts, partial fractions or trigonometric identities to evaluate exact integrals and areas with correctly transformed bounds.

Exam points

recognise the integrand structure and choose substitution, parts or partial fractions
rewrite trigonometric or rational expressions into standard integrable terms
transform bounds with the substitution and simplify the definite result into the required exact form

Covered syllabus points

3.5.1—Extended integration rules 3.5.2—Trig integration 3.5.3—Partial fractions 3.5.4—An integrand of the form x 3.5.5—Inverse tangent 3.5.6—Integration by substitution

[Maximum number: 4]

Hence show that $\int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi}(\operatorname{cosec} 2 \theta-\cot 2 \theta) \mathrm{d} \theta=\frac{1}{2} \ln 2$ .

[Maximum number: 5]

Given that $\int_{a}^{a+14} \frac{1}{3 x} \mathrm{~d} x=\ln 2$ , find the value of the positive constant a.

[Maximum number: 6]

Hence find the exact value of $\int_{-\frac{1}{8} \pi}^{\frac{1}{8} \pi}\left(\cos ^{4} \theta-\sin ^{4} \theta+4 \sin ^{2} \theta \cos ^{2} \theta\right) \mathrm{d} \theta$ .

[Maximum number: 6]

Find the exact value of $\int_{0}^{0.5} x \tan ^{-1}(2 x) \mathrm{d} x$ .

[Maximum number: 4]

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

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

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

Exam points

Covered syllabus points

Question 6(b)

Question 4

Question 7(b)

Question 10(b)

Question 9(b)