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Integration By Substitution

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