CAIE A-Level Mathematics 2.5.2 Trig integration

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

Exam points

rewrite sin^2 x or cos^2(2x) using double-angle formulae before integrating
apply exact limits involving π to obtain values with radicals, fractions or π terms

[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.

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2.5.2—Trig integration question 1