CAIE A-Level Mathematics 2.5 Integration Question Bank

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

Practise integrating exponential, logarithmic, rational and trigonometric forms to find exact values, areas and approximations.

Exam points

integrate e^(ax+b) or 1/(ax+b), then apply limits and simplify logarithms exactly
use trig identities before integrating powers of sin or cos over stated limits
apply the trapezium rule with given ordinates and judge over- or under-estimates from the curve

Covered syllabus points

2.5.1—Extended integration rules 2.5.2—Trig integration 2.5.3—Trapezium rule

Figure for Question 4(b) — CAIE A-Level Mathematics

The diagram shows the curve with equation $y=6 \mathrm{e}^{2 x}-\mathrm{e}^{3 x}$ . The shaded region is bounded by the axes and the curve.

Find the area of the shaded region. Give your answer in the form $\frac{p}{q}$ , where p and q are integers.

[Maximum number: 7]

Show that $\int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi}\left(4 \cos ^{2} 2 x+\frac{1}{\cos ^{2} x}\right) d x=\frac{3}{4} \sqrt{3}+\frac{1}{6} \pi-1$ .

[Maximum number: 6]

The diagram shows the curves with equations $y=\sqrt[3]{5 x^{2}+7}$ and $y=\frac{27}{2 x+5}$ for $x \geqslant 0$ .
The curves meet at the point (2,3).
Region A is bounded by the curve $y=\sqrt[3]{5 x^{2}+7}$ and the straight lines x=0, x=2 and y=0.
Region B is bounded by the two curves and the straight line x=0.

(a)

Use the trapezium rule with two intervals to find an approximation to the area of region A. Give your answer correct to 3 significant figures.

[ 3 ]

(b)

Deduce an approximation to the area of region B. Give your answer correct to 3 significant figures.

[ 1 ]

(c)

State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region B.

[ 2 ]

Exam points

Covered syllabus points

Question 4(b)

Question 6

Question 6

Question 6(a)

Question 6(c)

Question 6(d)