CAIE A-Level Mathematics 2.5 Integration Question Bank

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

[Maximum number: 5]

Figure for Question 2.5 Integration question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Show that the x-coordinate of B is $\ln 2$ and hence find the area of the shaded region.

Attempt to find x-coordinate of B

M1

$8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}=0$ .

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$

A1

AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$

B1

Use limits 0 and $\ln 2$ correctly to find area

M1

For integral of form $k_{3} \mathrm{e}^{-x}+k_{4} \mathrm{e}^{2 x}$ where $k_{3} k_{4} \neq 0$ .
$k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $\frac{5}{2}$

A1

OE

[Maximum number: 2]

The expression $\mathrm{f}(\theta)$ is defined by $\mathrm{f}(\theta)=12 \sin \theta \cos \theta+16 \cos ^{2} \theta$ .

Find $\int f(\theta) d \theta$ .

Integrate trigonometry term from part (a) to obtain $k_{1} \sin (2 \theta$ -their 0.6435)

M1

any non-zero constant $k_{1}$ .

Obtain $5 \sin (2 \theta-0.6435)+8 \theta$

A1

condone absence of ... +c.

Alternative method for question 8(c)

Integrate to obtain at least form $k_{2} \cos 2 \theta+k_{3} \sin 2 \theta$

M1

any non-zero constants $k_{2}, k_{3}$ .

Obtain $-3 \cos 2 \theta+4 \sin 2 \theta+8 \theta$

A1

condone absence of ... +c.

[Maximum number: 3]

Figure for Question 2.5 Integration question 3 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\frac{2 \ln x}{3 x+1}$ . The curve crosses the x-axis at the point A and has a maximum point B. The shaded region is bounded by the curve and the lines x=3 and y=0.

Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 decimal places.

Use y-values [0], $\frac{2}{7} \ln 2$ or 0.1980 and $\frac{2}{10} \ln 3$ or 0.2197

B1

Use correct formula, or equivalent, with h=1

M1

Obtain 0.31

A1

All question bank results loaded

Covered syllabus points

2.5 Integration question 1

2.5 Integration question 2

2.5 Integration question 3