Integrals of Composite Functions (College Board AP® Calculus AB)

Integrating composite functions

What is meant by integrating composite functions?

• Integrating composite functions refers to integrating 'by inspection'

  • by spotting that the chain rule would be used in the inverse (differentiation) process

  • This is sometimes referred to as 'reverse chain rule'

• This method can be used to integrate the product of

  • a composite function

  • and the derivative of its secondary ('inside') function

• In function notation, this method is to integrate integrals of the form

  • By the chain rule,

  • And differentiation and integration are inverse operations, therefore

    • 

• If coefficients do not match exactly, ‘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

      • 

Special case: a function raised to a power

• One common example is an integral involving a function raised to a power

• In this case the general pattern becomes

  • 

What are the steps for integrating composite functions?

• STEP 1
  Spot the ‘main’ function

  • E.g. 

  • Think: "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.

    • Now is the exact derivative of

  • So

• After some practice, you may find Step 2 is not needed (because you can do it in your head)

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

Integrating f'(x)/f(x)

How do I integrate f'(x)/f(x) ?

• A particularly useful special case of integrating composite functions is

  • 

    • I.e.  the numerator of a fraction being integrated is the derivative of the denominator

  • Make sure you recognise this pattern!

    • It speeds up and simplifies integrals of this sort

• 'Adjust and compensate' may need to be used to deal with any coefficients

  • e.g.