E6, H2PureStretch

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.

30 min Integration

The big picture

Powers such as $\sin^2x$ and $\cos^2x$ are not handled by dividing by a new power in x. Rewrite them as $(1-\cos2x)/2$ or $(1+\cos2x)/2$ . For products $\sin x\cos x$ , use $\sin2x=2\sin x\cos x$ . The trig argument still has an inner coefficient, so reverse the chain rule after applying the identity. Keep angle units in radians. With $\cos^2(3x)$ the power-reduction identity produces $\cos6x$ , so its integral has a factor of $1/6$ . Setting $u=\cos x$ without the accompanying sine factor does not produce a direct power integral; inspect the complete differential rather than just the visible square.

Edexcel A level Maths: 11.3 Using Trigonometric Identities To IntegrateCurated by Kepler, not produced by us. Watch it, then work through 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 $\cos^2 x = \tfrac12(1 + \cos 2x)$ integrates easily. Similarly $\sin^2 x = \tfrac12(1 - \cos 2x)$ .

\cos^2 x = \tfrac12(1 + \cos 2x), \quad \sin^2 x = \tfrac12(1 - \cos 2x)

Rewrite before integrating.

Example 1Find

\int \cos^2 x\,dx

= \int \tfrac12(1 + \cos 2x)\,dx

= \tfrac12 x + \tfrac14\sin 2x + c

Answer

\tfrac12 x + \tfrac14\sin 2x + c

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

Use a double-angle identity before integrating

Powers such as $\sin^2x$ and $\cos^2x$ are not handled by dividing by a new power in x. Rewrite them as $(1-\cos2x)/2$ or $(1+\cos2x)/2$ .

For products $\sin x\cos x$ , use $\sin2x=2\sin x\cos x$ . The trig argument still has an inner coefficient, so reverse the chain rule after applying the identity.

Keep angle units in radians. With $\cos^2(3x)$ the power-reduction identity produces $\cos6x$ , so its integral has a factor of $1/6$ . Setting $u=\cos x$ without the accompanying sine factor does not produce a direct power integral; inspect the complete differential rather than just the visible square.

Example 1Evaluate

\int_0^{\pi/2}\cos^2x\,dx

1Use

\cos^2x=(1+\cos2x)/2

2The antiderivative is

x/2+\sin2x/4

3At

\pi/2

, this is

\pi/4

; at 0, it is zero.

Answer

\pi/4

. The average value of

\cos^2x

on this interval is

1/2

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

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 results to remember

\int \cos^2 x\,dx = \tfrac12 x + \tfrac14\sin 2x + c

Via cos² identity.

\int \sin^2 x\,dx = \tfrac12 x - \tfrac14\sin 2x + c

Via sin² identity.

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).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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