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Techniques of Integration (DP IB Maths: AI HL)
Revision Note
Integrating Composite Functions (ax+b)
What is a composite function?
- A composite function involves one function being applied after another
- A composite function may be described as a “function of a function”
- This Revision Note focuses on one of the functions being linear – i.e. of the form
How do I integrate linear (ax+b) functions?
- A linear function (of) is of the form
- The special cases for trigonometric functions and exponential and logarithm functions are
- There is one more special case
- where
- , in all cases, is the constant of integration
- All the above can be deduced using reverse chain rule
- However, spotting them can make solutions more efficient
Examiner Tip
- Although the specific formulae in this revision note are NOT in the formula booklet
- almost all of the information you will need to apply reverse chain rule is provided
- make sure you have the formula booklet open at the right page(s) and practice using it
Worked example
Find the following integrals
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Reverse Chain Rule
What is reverse chain rule?
- The Chain Rule is a way of differentiating two (or more) functions
- Reverse Chain Rule (RCR) refers to integrating by inspection
- spotting that chain rule would be used in the reverse (differentiating) process
How do I know when to use reverse chain rule?
- Reverse chain rule is used when we have the product of a composite function and the derivative of its secondary function
- Integration is trickier than differentiation; many of the shortcuts do not work
- For example, in general
- However, this result is true if is linear
- Formally, in function notation, reverse chain rule is used for integrands of the form
-
- this does not have to be strictly true, but ‘algebraically’ it should be
- if coefficients do not match ‘adjust and compensate’ can be used
- e.g. is not quite the derivative of
- the algebraic part is 'correct'
- but the coefficient 5 is ‘wrong’
- use ‘adjust and compensate’ to ‘correct’ it
- this does not have to be strictly true, but ‘algebraically’ it should be
- A particularly useful instance of reverse chain rule to recognise is
-
- i.e. the numerator is (almost) the derivative of the denominator
- 'adjust and compensate' may need to be used to deal with any coefficients
- e.g.
How do I integrate using reverse chain rule?
- If the product can be identified, the integration can be done “by inspection”
- there may be some “adjusting and compensating” to do
- Notice a lot of the "adjust and compensate method” happens mentally
- this is indicated in the steps below by quote marks
- Differentiation can be used as a means of checking the final answer
- After some practice, you may find Step 2 is not needed
- Do use it on more awkward questions (negatives and fractions!)
- If the product cannot easily be identified, use substitution
Examiner Tip
- Before the exam, practice this until you are confident with the pattern and do not need to worry about the formula or steps anymore
- This will save time in the exam
- You can always check your work by differentiating, if you have time
Worked example
A curve has the gradient function.
Find an expression for.
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Substitution: Reverse Chain Rule
What is integration by substitution?
- When reverse chain rule is difficult to spot or awkward to use then integration by substitution can be used
- substitution simplifies the integral by defining an alternative variable (usually) in terms of the original variable (usually)
- everything (including “” and limits for definite integrals) is then substituted which makes the integration much easier
How do I integrate using substitution?
STEP 1
Identify the substitution to be used – it will be the secondary function in the composite function
So in and
STEP 2
Differentiate the substitution and rearrange
can be treated like a fraction
(i.e. “multiply by” to get rid of fractions)
- For definite integrals, a GDC should be able to process the integral without the need for a substitution
- be clear about whether working is required or not in a question
Examiner Tip
- Use your GDC to check the value of a definite integral, even in cases where working needs to be shown
Worked example
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