Evaluate the following integral for :
Since we know
. Then we write,
Now we consider two cases, and
. First, the case
. Then,
(since
and
must have the same sign by this exercise, Section I.3.5, Exercise #4). Therefore,
exists in
. Thus,
Now, make the substitution and
. This gives us,
Now, for the case we have
and so
. Therefore,
Again, we make a substitution, this time let and so
. Therefore, we have
, and so
Now, we simplify and replace with
to obtain