Integrating with Trigonometric Identities (Edexcel IAL Maths)

Spec point — 5.2

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 trigonometric 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
  • sin²x, cos²x, tan²x, cosec²x, sec²x, cot²x all appear in identities

How do I integrate tan²x, cot²x, sec²x and cosec²x?

• 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 expressions involving sin x and cos x?

• 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

• sinⁿkx cos kx or sin kx cosⁿkx 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 x?

• This is a standard result from the formulae booklet

How do I integrate other expressions involving 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

Exam Hint

• 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