When we use integration by parts, we have a choice for
and
In this problem, we can either let
and
or let
and
While there is not a universal rule for how to choose
and
a good guideline is this: do so in a way that
is at least as simple as the original problem
This leads us to choose and from which it follows that and With this substitution, the rule for integration by parts tells us that
All that remains to do is evaluate the (simpler) integral Doing so, we find
Observe that when we get to the final stage of evaluating the last remaining antiderivative, it is at this step that we include the integration constant,