Trigonometric identities in integration | Year 12

01 / LEARN

The idea, step by step

Make the integrand integrable

Power-reduction identities give ${sin}^{2} x = (1 - cos 2 x) / 2$ and ${cos}^{2} x = (1 + cos 2 x) / 2$ . Integrate the rewritten sum and remember the inner frequency in the antiderivative.

For an odd sine power, keep one sin x factor and replace the remaining sin squared using 1-cos squared. Then u=cos x absorbs the single sine factor, with a negative differential. A similar strategy works for odd cosine powers.

Select an identity for the task

Products can be rewritten as sums, for example $sin x cos x = sin 2 x / 2$ . When two different frequencies occur, a product-to-sum identity may produce two simple terms. Check signs by testing a convenient angle.

Do not replace a squared sine by half a sine. An identity must be true for every x in the relevant domain. Differentiate the final antiderivative and simplify with the same identity to verify equivalence.

A common trap

The antiderivative of sin squared is obtained by applying the power rule directly to sin.

Instead: The power rule applies to powers of the integration variable; a composite expression needs a chain-rule factor or an identity.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Find $\int {sin}^{2} x d x$ .

Use ${sin}^{2} x = (1 - cos 2 x) / 2$ .
Integrate the constant and cosine terms separately.

x / 2 - sin 2 x / 4 + C

Example 2

Worked example 2

Find $\int {sin}^{3} x d x$ .

Write the integrand as $(1 - {cos}^{2} x) sin x$ .
Use u=cos x, so du=-sin x dx.

- cos x + {cos}^{3} x / 3 + C

Check your understanding

Rewrite sin squared x times cos squared x.

Show answer

$(1 - cos 4 x) / 8$ .

Use sin x cos x=sin 2x/2, then reduce the square.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $\int {cos}^{2} (2 x) d x$ .
2 marks
Find $\int sin x cos x d x$ .
2 marks
Find $\int {sin}^{2} x {cos}^{2} x d x$ .
2 marks
Evaluate $\int_{0}^{π / 2} {sin}^{2} x d x$ .
2 marks
Find $\int sin x cos (3 x) d x$ .
2 marks
Find $\int {cos}^{3} x d x$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $\int {cos}^{2} (2 x) d x$ .
2 marks
Use $(1 + cos 4 x) / 2$ .
Divide the cosine antiderivative by its frequency 4.
$x / 2 + sin 4 x / 8 + C$ .
Find $\int sin x cos x d x$ .
2 marks
Use u=sin x.
Integrate u du.
${sin}^{2} x / 2 + C$ .
Find $\int {sin}^{2} x {cos}^{2} x d x$ .
2 marks
Rewrite as $(1 - cos 4 x) / 8$ .
Integrate each term.
$x / 8 - sin 4 x / 32 + C$ .
Evaluate $\int_{0}^{π / 2} {sin}^{2} x d x$ .
2 marks
Use $x / 2 - sin 2 x / 4$ .
The sine endpoint terms are zero.
$π / 4$ .
Find $\int sin x cos (3 x) d x$ .
2 marks
Use $sin x cos 3 x = (sin 4 x - sin 2 x) / 2$ .
Integrate both terms with their frequencies.
$- cos 4 x / 8 + cos 2 x / 4 + C$ .
Find $\int {cos}^{3} x d x$ .
2 marks
Write $(1 - {sin}^{2} x) cos x$ .
Use u=sin x.
$sin x - {sin}^{3} x / 3 + C$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics integration techniques and applications, including inverse-trigonometric patterns, substitution, parts, simple partial fractions, areas and solids of revolution.

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