Techniques of Integration (DP IB Applications & Interpretation (AI)): 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

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

• 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 

STEP 1

Spot the ‘main’ function

e.g. 

"the main function is which would come from”

 STEP 2

‘Adjust’ and ‘compensate’ any coefficients required in the integral

e.g.  " would differentiate to "

“chain rule says multiply by the derivative of, which is”

“there is no '7' or ‘10’ in the integrand so adjust and compensate”

 STEP 3

Integrate and simplify

e.g. 

• 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

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)

STEP 3

Replace all parts of the integral

All terms should be replaced with equivalent terms, including

If finding a definite integral change the limits from-values to-values too

STEP 4

Integrate and either

substitute back in

or

evaluate the definte integral using the limits (either using a GDC or manually)

STEP 5

Find, the constant of integration, if needed

• 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