Integrating with Trigonometric Identities (Cambridge International A-Level Mathematics)

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

• You may need to use the compound angle formulae or the double angle formulae
• Note the difference between the ± and ∓ symbols!
• $\mathrm{sin}(A\pm B)\equiv\mathrm{sin}A\mathrm{cos}B\pm\mathrm{cos}A\mathrm{sin}B$
• $\cos ( A \pm B ) \equiv \cos A \cos B --or-+ \sin A \sin B$
• $\tan ( A \pm B ) \equiv \frac{\tan A \pm \tan B}{1 --or-+ \tan A \tan B}$ $(A\pm B\neq(k+\frac{1}{2})π)$
• $\mathrm{sin}2A\equiv2\mathrm{sin}A\mathrm{cos}A$
• $\mathrm{cos}2A\equiv\mathrm{cos}^{2}A-\mathrm{sin}^{2}A\equiv2\mathrm{cos}^{2}A-1\equiv1-2\mathrm{sin}^{2}A$
• $\mathrm{tan}2A\equiv\frac{2\mathrm{tan}A}{1-\mathrm{tan}^{2}A}$

How do I know which trigonometric 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
• sin²x, cos²x, tan²x, cosec²x, sec²x, tan²x all appear in identities
• This is a matter of experience, familiarity and recognition

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 with 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?

• $\int \tan x d x = \ln \left| \sec x \right| + c$

• This is not in the formula booklet

• It can be derived from writing $\mathrm{tan}x$ as $\frac{\mathrm{sin}x}{\mathrm{cos}x}$ and recognising that $\int \frac{f ' ( x )}{f ( x )} d x = \ln \left| f ( x ) \right|$

  • Note that this is in the formula booklet

How do I integrate other expressions with 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

Worked Example

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.