medium difficulty6 min

Integration Using Trigonometric Identities

Some trigonometric integrals are first simplified using identities, such as sin²x=(1-cos2x)/2 and cos²x=(1+cos2x)/2, before applying standard integration formulas.

In one line: First reduce the trig expression, then integrate the reduced form.

Practice This Concept

Understand it

Direct integration may not be possible when powers or products of trigonometric functions appear. Identities convert the integrand into sums or standard forms. The chosen identity must match the power, product, or angle structure in the question.

Example

∫sin²x dx=∫(1-cos2x)/2 dx=x/2-sin2x/4+C.

Formula and rule

Useful identities: sin²x=(1-cos2x)/2, cos²x=(1+cos2x)/2, 2sinxcosx=sin2x, sin²x+cos²x=1, 1+tan²x=sec²x.

Apply identities before integration when the original form is not standard.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate ∫sinx cosx dx.

1Use 2sinxcosx=sin2x, so sinxcosx=(1/2)sin2x.
2Integral =(1/2)∫sin2x dx=(1/2)(-cos2x/2)+C=-cos2x/4+C.
3Equivalent answer: sin²x/2+C.

Method you can reuse

1Inspect whether the integrand is a square, product, or expression involving tan and sec.
2Choose a matching identity.
3Rewrite the full integrand.
4Integrate term-wise.
5Check angle multipliers such as 2x or ax.
6Add +C when indefinite.

Visual tip

No diagram is needed. The essential support is a list of identities and careful angle handling, especially when 2x appears.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find ∫cos²x dx.

Think first, then show answer

Using cos²x=(1+cos2x)/2, integral =x/2+sin2x/4+C.

Common wrong answer

Writing ∫cos2x dx=sin2x+C instead of sin2x/2+C, missing the factor from the derivative of 2x.

Don't mix this up

Students sometimes integrate sin²x as if it were sin x, which is incorrect because powers of trigonometric functions need reduction or another suitable identity.

Exam tip

Before integrating a trigonometric power or product, ask whether an identity can reduce it to first powers or a standard derivative pair.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask direct evaluation, simplification before integration, or comparison of equivalent answers. Long-answer questions may combine identities with substitution.

1
Enough for 1 mark
Some trigonometric integrals are first simplified using identities, such as sin²x=(1-cos2x)/2 and cos²x=(1+cos2x)/2, before applying standard integration formulas.
2
Add for 2 marks
Direct integration may not be possible when powers or products of trigonometric functions appear.
3
Add for 3 marks
Identities convert the integrand into sums or standard forms.

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