OCR A Level Maths: Pure

Revision Notes

8.2.7 Integrating with Trigonometric Identities

Integrating with Trigonometric Identities

What are trigonometric identities?

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

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

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

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

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

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

  • 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!

Notes sin_cos_powers, AS & A Level Maths revision notes 

  • 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

 

Notes sin_cos_squared, AS & A Level Maths revision notes

How do I integrate tan?

Notes int_tan, AS & A Level Maths revision notes

 

  • 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) !

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

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

Example soltn_a, AS & A Level Maths revision notesExample soltn_b, AS & A Level Maths revision notes

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Paul

Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.