CAIE A-Level Mathematics AS 2.5 Integration Questions

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

Obtain $\mathrm{e}^{3 x}=8$ and hence $x=\ln 2$
AG so necessary detail needed.

A0 if decimals used.

Integrate to obtain $-8 \mathrm{e}^{-x}-\frac{1}{2} \mathrm{e}^{2 x}$
Use limits 0 and $\ln 2$ correctly to find area
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 $\frac{5}{2}$
OE

[Maximum number: 7]

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

Express $4 \cos ^{2} 2 x$ in the form $k_{1} \cos 4 x+k_{2}$
where $k_{1} k_{2} \neq 0$ .

Obtain correct $2 \cos 4 x+2$
Marking guidance:

Allow unsimplified.

State or imply $\frac{1}{\cos ^{2} x}=\sec ^{2} x$
Maybe implied by integration.

Integrate to obtain $k_{3} \sin 4 x+k_{4} x+\tan x$

*M1

where $k_{3} k_{4} \neq 0$ .

Obtain correctly $\frac{1}{2} \sin 4 x+2 x+\tan x$
Use limits correctly with correct values of $\sin \frac{4}{3} \pi$ and $\tan \frac{1}{3} \pi$ indicated
Confirm given result $\frac{3}{4} \sqrt{3}+\frac{1}{6} \pi-1$ with sufficient detail
AG

7

[Maximum number: 3]

Figure for Question 3 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=\sqrt{\sin 2 x+\sin ^{2} 2 x}$ for $0 \leqslant x \leqslant \frac{1}{6} \pi$ . The shaded region is bounded by the curve and the straight lines $x=\frac{1}{6} \pi$ 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 significant figures.

Use y-values ( 0 ), $\quad \sqrt{\sin \frac{1}{6} \pi+\sin ^{2} \frac{1}{6} \pi}, \quad \sqrt{\sin \frac{1}{3} \pi+\sin ^{2} \frac{1}{3} \pi}$ or decimal equivalents
(0), $\sqrt{0.75}$ or 0.866, $\sqrt{1.616}$ or 1.271.

Use correct formula, or equivalent, with $h=\frac{1}{12} \pi$
Must be using ' y ' values.

May do as 2 separate trapezia ( 0.113359+ 0.27976).

Obtain 0.39
Marking guidance:

Allow 0.393 but not greater accuracy.

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Covered syllabus points

Question 1

Question 2

Question 3