Trying to understand this problem...Integration by Parts

I have been racking my head for the past week trying to understand these types of exercises, and for the most part, they're usually understandable as I DO study. However, I have hit a major speed bump along the way. I can't for the life of me do the following exercise. I have searched, read, studied the procedures over and over again and can't ever seem to arrive to the answer provided in the answer sheet.

Solve for x.

Answer:

I don't need the answer, because I already have it. What I'm trying to understand is HOW exactly they arrive to the answer. I've consulted the internet, and even my book on Calculus for Dummies. I have a test coming up next thursday (October 22, 2009...NEXT thursday, not THIS thursday) and even though I have plenty of time till then, I'd rather understand this now under calm conditions rather than wait for the panic attack to set in. So can anyone help me understand how to get from point A to point B.

If anyone could be kind enough as to explain to me how to do this step-by-step I would be VERY grateful.

Thank you so much in advance!