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AQA A Level Maths: Pure复习笔记8.2.7 Integrating with Trigonometric Identities
Category:
A-level课程
,
教材笔记
,
福利干货
Date: 2022年8月9日 上午11:21
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 trig 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
sin
2
x, cos
2
x, tan2 x, cosec
2
x, sec
2
x, tan
2
x all appear in identities
How do I integrate tan
2
, cot
2
, sec
2
and cosec
2
?
The integral of sec
2
x is tan x (+c)
This is because the derivative of tan x is sec
2
x
The integral of cosec
2
x is -cot x (+c)
This is because the derivative of cot x is -cosec
2
x
The integral of tan
2
x can be found by using the identity to rewrite tan
2
x before integrating:
1 + tan
2
x = sec
2
x
The integral of cot
2
x can be found by using the identity to rewrite cot
2
x before integrating:
1 + cot
2
x = cosec
2
x
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
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!
sin
2
kx and cos
2
kx can be integrated by using the identity for cos 2A
For sin
2
A, cos 2A = 1 - 2sin
2
A
For cos
2
A, cos 2A = 2cos
2
A – 1
How do I integrate tan?
This is a standard result from the formulae 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) !
Experience, familiarity and recognition are important – practice, practice, practice!
Problem-solving techniques
Exam 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.
Worked Example
转载自savemyexams
Previous post: AQA A Level Maths: Pure复习笔记8.2.6 Harder Substitution
Next post: AQA A Level Maths: Pure复习笔记8.2.8 Integration by Parts
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