Since no one has explicitly come out and said this: you basically can't do this integral. It isn't that the integral doesn't exist, it's that you can't express it as a so-called elementary function. Essentially, all your integration by parts and substitution tricks are completely useless.
Don't worry about getting asked this on an exam. Most instructors don't ask impossible questions. The most one could reasonably expect from a student is to show that . This isn't so bad if given the appropriate hint, but the usual way of solving it requires multivariate calculus, although you can do without that I think.
dwsmith did, in fact, "come out and say" that it was NOT an elementary integral. He did not say that "you basically can't do this integral" because you can. You just have to use the non-elementary error function, erf(x), just as he showed.
slapmaxwell1, you said "..i used maxima and got the same answer; but how would i get the answer by hand? with paper and pencil?"
Well, what was the problem? You can't "use maxima and minima" to get an integral, which was the problem you originally posed. What problem are you really trying to solve?
Maxima is a computer-algebra system. He plugged the integral into that; he wasn't saying he was trying to use optimization techniques to get the solution.
Anyways, I don't really want to get into an argument over this, but I don't really consider using erf(x) as "doing" the integral in the sense that he would be expected to do it if it was an assigned problem in a calculus class. OP doesn't nessecarily know what an elementary function is (and still might not really have a grasp on it after reading the wiki page), and I thought he would benefit from knowing that, in the sense that he is probably attempting to solve the problem, his attempts are futile, which wasn't really said explicitly.