From theorem 5.9 about the derivative of the composition of two functions, we deduce the method of integration by substitution (i.e. change of variable).
Let and be two functions, each of them being differentiable on its own domain. By Theorem 5.9, is differentiable and we have . It follows that:
Denote ; then and Equation 5 can be written in the following form:
Denote . We have:
Now let ; then , i.e. . We have:
We have: , thus:
Denote , thus and we have:
Noah Dana-Picard 2007-12-28