Reverse Chain Rule (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”

  • E.g.

    • First the function is applied to

    • Then the function is applied to

• 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, where and are constants

• The special cases for trigonometric functions and exponential and logarithm functions are

  •  

  •  

  • 

  • 

• There is one more special case you should know

  •  where 

• , in all cases, is the constant of integration

• All the above can be deduced using reverse chain rule

  • However, recognising the patterns and knowing these results can make solutions more efficient

Reverse Chain Rule

What is reverse chain rule?

• The Chain Rule is a way of differentiating composite functions made up of two (or more) functions

• Reverse Chain Rule (RCR) refers to integrating by inspection

  • You do this by 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

  • E.g.

    • is a composite function

      • is the primary (or main or 'outside') function

      • is the secondary (or 'inside') function

    • is the derivative of the secondary function

• Formally, in function notation, reverse chain rule can be written in the following way 

  
  

  • E.g.

    • , , and

    • And is the derivative of , so

    • Therefore

• Be sure also to recognise this useful instance of reverse chain rule

  
  
  

  • I.e.  the numerator is the derivative of the denominator

  • E.g.

    • 

    • And the derivative of that is

    • Therefore

How do I integrate using reverse chain rule?

• If you can spot the patterns, the integration can be done “by inspection”

  • Though there may be some “adjusting and compensating” to do

• A lot of the 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

• After some practice, you may find Step 2 is not needed

  • Do use it on more awkward questions (negatives and fractions!)