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.