17.4 Integration by Substitution

We will now use the chain rule to find some antiderivatives. Let be a real
valued function that is continuous on some interval and differentiable on
the
interior of , and let be a function such that has an antiderivative
on some interval . We will suppose that and
. It then follows that is continuous on
and differentiable on
, and

There is a standard ritual for using (17.33) to find when an antiderivative can be found for . We write:

Let . Then (or
), so

Let . Then , so

Suppose is a function on an interval such that is never zero on
the interior of , and suppose that is an inverse function for . Then

for all in the interior of , so

We now apply the ritual (17.34): Let . Then , so

If we can find an antiderivative for , then

We have shown that if is an inverse function for , then

There is a ritual associated with this result also. To find :

Let . Then so .

Hence

Let . Then so .

Thus

We can now use integration by parts to find . Let

Then

Hence

Let . Then so .

Let . Then so .

Hence