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

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!

• $\sin (A\pm B)\equiv \sin A\cos B\pm \cos A\sin B$

• $\cos (A\pm B)\equiv \cos A\cos B\mp \sin A\sin B$

• $\tan (A\pm B)\equiv \frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$  $(A\pm B\ne (k+\frac{1}{2}\left)\mathrm{\pi }\right)$

• $\sin 2A\equiv 2\sin A\cos A$

• $\cos 2A\equiv {\cos }^{2}A-{\sin }^{2}A\equiv 2{\cos }^{2}A-1\equiv 1-2{\sin }^{2}A$

• $\tan 2A\equiv \frac{2\tan A}{1-{\tan }^{2}A}$

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

• 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

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

• 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!

• 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?

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

• This is not in the formula booklet

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

  • 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