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

  • 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 in and

STEP 2
Differentiate the substitution and rearrange

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

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

STEP 4

Integrate and either

substitute back in

or

evaluate the definte integral using the limits (either using a GDC or manually)

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 working is required or not in a question