# Trigonometric integrals

Use identities to turn powers of sine and cosine into forms you can integrate. For example, sin^2 x = (1 - cos 2x)/2.

Maths (university) · Part 06: Integration techniques and uses · Card 4 of 9 · Last updated 2026-10-19

## Analogy

Changing a hard-to-read photo into a clearer one before you study it.

## Example

**Integral of sin^2 x from 0 to pi**

sin^2 x = (1 - cos 2x)/2.
Antiderivative: x/2 - (sin 2x)/4.
At pi: pi/2. At 0: 0.
Result: pi/2.

## In practice

Needed in physics and engineering for waves and rotating motion.

## Practice

### Flashcards

- **Which identity helps integrate sin^2 x?** sin^2 x = (1 - cos 2x)/2.

## Questions people ask

### Which identity helps with sine squared and cosine squared?

The double-angle identities: sin squared x equals one half times one minus cos 2x, and cos squared x equals one half times one plus cos 2x. They replace a square with a single cosine that you can integrate directly. For example, sine squared over 0 to pi gives pi over 2, a result you can check.

### Why is the integral of sine squared not just sine cubed over 3?

That rule only works when the derivative of the inner function also appears, as in substitution. Here sin squared x has no cos x factor next to it. Without that, the power rule for integrals does not apply, so you need an identity. Without it, differentiating your wrong answer would show the mistake straight away.

## Related concepts

- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)
- [Choosing u and dv](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-2)
- [Partial fractions](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-3)
- [Integration by substitution](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-7)

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