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Integrating with Trigonometric Identities (Cambridge (CIE) A Level Maths): Revision Note

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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 2 tan 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 identities

  • This 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 Tips and Tricks

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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