Integration by substitution
In calculus, integration by substitution is a method of evaluating an antiderivative or a definite integral by applying a change of variables. It is the integral counterpart of the chain rule for differentiation. For a definite integral, it can be shown as follows:
Where such that .
- Let a variable equal part of the integrand, so that its derivative will cancel with the other part of the integrand
- Apply the substitution
- Evaluate the integral in terms of the new variable
Definite integral example change
Consider the integral
Let to obtain and . In this case, the variable is not present, so the 1/3 can be factored out of the integrand. Since the integral is now in terms of , the bounds of integration (1 and 3 in this case), must be plugged in to the substitution u=3x. So the new bounds of integration are 3 and 9 to obtain,
The antiderivative of may also be found using integration by substitution and ends up being .