Integrating with Trigonometric Identities

What are trigonometric identities?

You should be familiar with the trigonometric identities
Make sure you can find them in the formula booklet

Notes trig_id_essential, A Level & AS Level Pure Maths Revision Notes

You may need to use the compound angle formulae or the double angle formulae
Note the difference between the ± and ∓ symbols!
- $\sin (A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$
- $\cos (A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$
- $\tan (A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ $(A \pm B \neq (k + \frac{1}{2}) π)$
- $\sin 2 A \equiv 2 \sin A \cos A$
- $\cos 2 A \equiv \cos^{2} A - \sin^{2} A \equiv 2 \cos^{2} A - 1 \equiv 1 - 2 \sin^{2} A$
- $\tan 2 A \equiv \frac{\tan 2 A}{1 - \tan^{2} A}$

How do I know which trig identities to use?

There is no set method
- Practice as many questions as possible
- Be familiar with trigonometric functions that can be integrated easily
- Be familiar with common identities – especially squared terms
- sin²x, cos²x, tan²x, cosec²x, sec²x, tan²x all appear in identitiesThis is a matter of experience, familiarity and recognition

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How do I integrate tan², cot², sec² and cosec²?

The integral of sec²x is tan x (+c)
- This is because the derivative of tan x is sec²x
The integral of cosec²x is -cot x (+c)
- This is because the derivative of cot x is -cosec²x
The integral of tan²x can be found by using the identity to rewrite tan²x before integrating:
- 1 + tan²x = sec²x
The integral of cot²x can be found by using the identity to rewrite cot²x before integrating:
- 1 + cot²x = cosec²x

How do I integrate sin and cos?

For functions of the form sin kx, cos kx … see Integrating Other Functions
sin kx × cos kx can be integrated using the identity for sin 2A
- sin 2A = 2sinAcosA

Notes sin_cos, AS & A Level Maths revision notes

sinⁿkx cos kx or sin kx cosⁿkx can be integrated using reverse chain rule or substitution
Notice no identity is used here but it looks as though there should be!

Notes sin_cos_powers, AS & A Level Maths revision notes

sin²kx and cos²kx can be integrated by using the identity for cos 2A
- For sin²A, cos 2A = 1 - 2sin²A
- For cos²A, cos 2A = 2cos²A – 1

Notes sin_cos_squared, AS & A Level Maths revision notes

How do I integrate tan?

$\int \tan x d x = \ln |\sec x| + c$
This is not in the formula booklet
It can be derived from writing $\tan x$ as $\frac{\sin x}{\cos x}$ and recognising that $\int \frac{f' (x)}{f (x)} d x = \ln |f (x)|$
- - Note that this is in the formula booklet

How do I integrate other trig functions?

The formulae booklet lists many standard trigonometric derivatives and integrals
- Check both the “Differentiation” and “Integration” sections
- For integration using the "Differentiation" formulae, remember that the integral of f'(x) is f(x) !

Notes Diff-Int-fb, A Level & AS Level Pure Maths Revision Notes

Experience, familiarity and recognition are important – practice, practice, practice!
Problem-solving techniques

Notes eg, AS & A Level Maths revision notes

Worked example

5-1-2-int-with-trig-example-solution-part-1

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Examiner Tip

Make sure you have a copy of the formulae booklet during revision.Questions are likely to be split into (at least) two parts:

- The first part may be to show or prove an identity
- The second part may be the integration

If you cannot do the first part, use a given result to attempt the second part.

Integrating with Trigonometric Identities (CIE A Level Maths: Pure 3)

Revision Note