Some integrals look intractible. Some of these intractible integrals can be made simpler and transformed into an integral that can be evaluated by making a suitable change of variables. We start with an integral
and make the substitution
We get
Example: Find
Substitute
into the integral
Now we integrate the simplified integral:
We are not finished yet. The last step – assuming we have no limits – is to substitute back the original substitution so that we end up with a function of
In this case
so the final answer is
Example: Find
Substitute
into the integral:
Now we integrate the simplified integral:
We need to end up with a function of
hence
Example: Find
Substitute
into the integral:
Now we integrate the simplified integral:
We need to end up with a function of
hence
