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

Integration by Substitution

What is integration by substitution?

• Integration by substitution is used when 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 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
  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
  If limits are involved, use to change them from  values to  values 
  

  STEP 4
  Rewrite the integral so everything is in terms of  rather than 
  This is the step when it may become apparent that  is needed in terms of 
  

  STEP 5
  Integrate with respect to u and either rewrite in terms of  or apply the limits using their  values

• 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 can be treated like a fraction it should be appreciated that this is a ‘shortcut’ and the maths behind it is beyond the scope of the IB course