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A LevelMaths: Pure 3CIERevision Notes5. Integration5.1 Further Integration5.1.2 Integrating with Trigonometric Identities

Integrating with Trigonometric Identities (CIE A Level Maths: Pure 3)

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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

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

  • You may need to use the compound angle formulae or the double angle formulae
  • Note the difference between the ± and ∓ symbols!
    • sin left parenthesis A plus-or-minus B right parenthesis space identical to sin A cos B space plus-or-minus space cos A sin B
    • cos left parenthesis A plus-or-minus B right parenthesis space identical to cos A cos B space minus-or-plus space sin A sin B
    • tan left parenthesis A plus-or-minus B right parenthesis space identical to fraction numerator tan A space plus-or-minus space tan B over denominator 1 space minus-or-plus space tan A tan B end fraction  stretchy left parenthesis A plus-or-minus B space not equal to space stretchy left parenthesis k space plus space 1 half stretchy right parenthesis straight pi stretchy right parenthesis
    • sin 2 A space identical to 2 sin A cos A
    • cos 2 A space identical to space cos squared A space minus space sin squared A space identical to space 2 cos squared A space minus space 1 space identical to 1 space minus space 2 sin squared A
    • tan 2 A space identical to space fraction numerator tan 2 A over denominator 1 space minus space tan squared A end fraction

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
    • sin2 x, cos2 x, tan2 x, cosec2 x, sec2 x, tan2 x all appear in identitiesThis is a matter of experience, familiarity and recognition

5-1-2-int-with-trig-diagram-1-cropped-2

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?

  • integral tan space x space d x space equals space ln open vertical bar sec space x close vertical bar space plus space c
  • This is not in the formula booklet
  • It can be derived from writing tan space x as fraction numerator sin space x over denominator cos space x end fraction and recognising that integral space fraction numerator f apostrophe left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis end fraction d x space equals space ln open vertical bar space f left parenthesis x right parenthesis close vertical bar
      • 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) !

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

Worked example

5-1-2-int-with-trig-example-solution-part-1

5-1-2-int-with-trig-example-solution-part-2

Examiner 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.

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    Lucy

    Author: Lucy

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.