11.3PureStretch

Using Trigonometric Identities

Many trig expressions cannot be integrated directly. The trick is to rewrite them with an identity — especially the double-angle forms — into something with a standard integral.

24 min Integration

Edexcel A level Maths: 11.3 Using Trigonometric Identities To IntegrateCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Integrate sin²x and cos²x
Use double-angle identities
Rewrite products into integrable forms
Apply to definite integrals

Power-reduction

You cannot integrate $cos^{2} x$ directly, but $cos^{2} x = \frac{1}{2} (1 + cos 2 x)$ integrates easily. Similarly $sin^{2} x = \frac{1}{2} (1 - cos 2 x)$ .

cos^{2} x = \frac{1}{2} (1 + cos 2 x), sin^{2} x = \frac{1}{2} (1 - cos 2 x)

Rewrite before integrating.

= \int \frac{1}{2} (1 + cos 2 x) d x

= \frac{1}{2} x + \frac{1}{4} sin 2 x + c

Answer

\frac{1}{2} x + \frac{1}{4} sin 2 x + c

Tip — See a squared sin or cos? Reach for a double-angle identity first.

Formula recap

\int cos^{2} x d x = \frac{1}{2} x + \frac{1}{4} sin 2 x + c

Via cos² identity.

\int sin^{2} x d x = \frac{1}{2} x - \frac{1}{4} sin 2 x + c

Via sin² identity.

Common mistakes to avoid

Integrating cos²x to (cos³x)/3 type answers.

Use the identity first; there is no simple power rule for cos²x.

Forgetting the 1/a factor when integrating cos 2x.

∫cos 2x dx = ½ sin 2x.

Key takeaways

cos²x = ½(1+cos2x); sin²x = ½(1−cos2x).
Rewrite with an identity, then integrate term by term.
Mind the 1/a factor for cos(2x).

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