Integration by Substitution (DP IB Analysis & Approaches (AA)): Revision Note

Integration by substitution

What is integration by substitution?

• Integration by substitution is used with an integrand where reverse chain rule is either not obvious or is not spotted

  • In the latter case it is like a “back-up” method for reverse chain rule

How do I use integration by substitution?

• For instances where the substitution is not obvious it will be given in a question

  • e.g.  Find  using the substitution 

• Substitutions are usually of the form 

  • In some cases and other variations are more convenient

    • As these would not be obvious, they would be given to you in a question

  • If need be, this can be rearranged to find  in terms of 

• Integration by substitution then involves rewriting the integral, including “”, in terms of 
    

• STEP 1
  Name the integral to save rewriting it later ( is often used)
  Identify the given substitution 
   

• STEP 2
  Find and rearrange into the form such that (some of) the integral can be rewritten in terms of 
   

• STEP 3
  For a definite integral, use to change the integration limits from  values to  values 
   

• STEP 4
  Rewrite the integral so everything is in terms of  rather than 

  • This is the step where you may need to have  in terms of 

• STEP 5
  Integrate with respect to and either

  • rewrite in terms of  (for an indefinite integral)

  • or apply the limits using their  values (for a definite integral)

• For quotients the substitution usually involves the denominator

• It may be necessary to use ‘adjust and compensate’ to deal with any coefficients in the integrand

• Although is a derivative, you can treat it like an ordinary fraction when doing integration by substitution

  • This is a ‘shortcut’ that will get you to the correct answer