Integration by substitution, sometimes called changing the variable, is used when an integral cannot be integrated by standard means. The method involves changing the variable to make the integral into one that is easily recognisable and can be then integrated.

The integral in this example can be done by recognition but integration by substitution, although a longer method is an alternative.

Example 1

Evaluate the following:

  1. \int {x{{(3{x^2} - 1)}^4}} dx {\hphantom{aaa} \text{let}\hphantom{aa}}u = 3{x^2} - 1

Example 2

Evaluate the following:

  1. \int {x{{(2x + 3)}^4}} dx {\hphantom{aaa} \text{let}\hphantom{aa}}u = 2x + 3