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

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 instead

  • 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 if you are integrating something like, use the substitution

  • E.g.

    • Use the substitution

• STEP 2
  Differentiate the substitution and rearrange

  can be treated like a fraction (i.e. “multiply by” to get rid of fractions)

  • E.g.

• 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)

  • E.g.

• STEP 4

  Integrate and either substitute back in (indefinite integral) or evaluate using the limits (definite integral)

  • E.g.

• 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 or not working is required in a question