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