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:
formula_1
Where formula_2 such that formula_3.
Definite integral example.
Consider the integral
formula_4
Let formula_5 to obtain formula_6 and formula_7. In this case, the formula_8 variable is not present, so the 1/3 can be factored out of the integrand. Since the integral is now in terms of formula_9, 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,
formula_10
The antiderivative of formula_11 may also be found using integration by substitution and ends up being formula_12.