# CIE A Level Maths: Pure 3复习笔记5.1.2 Integrating with Trigonometric Identities

### 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 • You may need to use the compound angle formulae or the double angle formulae
• Note the difference between the ± and  symbols! #### 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
• sinxcosxtanxcosecxsecxtanx all appear in identitiesThis is a matter of experience, familiarity and recognition #### 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 tan2before integrating:
• 1 + tan2x = sec2x
• The integral of cot2x can be found by using the identity to rewrite cot2before integrating:
• 1 + cot2x = cosec2x

#### How do I integrate sin and cos?

• For functions of the form sin kxcos kx … see Integrating Other Functions
• sin kx × cos kx can be integrated using the identity for sin 2A
• sin 2A = 2sinAcosA • sinkx cos kx or sin kx coskx can be integrated using reverse chain rule or substitution
• Notice no identity is used here but it looks as though there should be! • sinkx and coskx can be integrated by using the identity for cos 2A
• For sinA, cos 2A = 1 - 2sinA
• For cosA, cos 2A = 2cosA – 1 #### How do I integrate tan? 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) ! • Experience, familiarity and recognition are important – practice, practice, practice!
• Problem-solving techniques #### Worked Example  #### Exam 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. 