CAIE A-Level Mathematics AS 2.5.2 Trig Integration Questions

Practise using trig relationships to rewrite integrands, integrate powers of sin and cos, and evaluate exact definite integrals.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[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

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CAIE A-Level Mathematics AS 2.5.2 Trig Integration Questions question 1