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

Integrating with Trigonometric Identities

What are trigonometric identities?

8.2.6-Notes-trig_id_essential

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

    8.2.6-Notes-trig_id_fb

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, tanx all appear in identities

8.2.6-Notes-squared_trig_id

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

8.2.6-Notes-sin_cos

  • 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!

8.2.6-Notes-sin_cos_powers

  • sinkx 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

8.2.6-Notes-sin_cos_squared

How do I integrate tan?

8.2.6-Notes-int_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) !

8.2.6-Notes-Diff-Int-fb

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

8.2.6-Notes-eg

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.

Worked Example

8.2.6-Example-soltn_a8.2.6-Example-soltn_b

 

 

 

 

 

 

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