AQA A Level Maths: Pure复习笔记8.2.7 Integrating with Trigonometric Identities

What are trigonometric identities?

• Some are given in the formulae booklet
  • Be sure to note the difference between the ± and ∓ symbols!

  

How do I know which trig identities to use?

• There is no set method
• This is a matter of experience, familiarity and recognition
  • Practice as many questions as possible
  • Be familiar with trigonometric functions that can be integrated easily
  • Be familiar with common identities – especially squared terms
  • sin² x, cos² x, tan2 x, cosec² x, sec² x, tan² x all appear in identities

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

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

• sin² kx and cos² kx can be integrated by using the identity for cos 2A
  • For sin² A, cos 2A = 1 - 2sin² A
  • For cos² A, cos 2A = 2cos² A – 1

How do I integrate tan?

• This is a standard result from the formulae 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