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 integraland make the substitutionWe get
Example: Find
Substituteinto 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 ofIn this caseso the final answer is
Example: Find
Substituteinto the integral:
Now we integrate the simplified integral:
We need to end up with a function ofhence
Example: Find
Substituteinto the integral:
Now we integrate the simplified integral:
We need to end up with a function ofhence