Exponential Integral - Integration by Parts with Inverse Functions

Can someone walk through this and let me know if everything is all right? I have a bad habit of making mistakes with my math and getting excited over nothing (Headbang).

You can find my work here: http://img9.imageshack.us/img9/4702/...279204d696.png

http://img9.imageshack.us/img9/4702/...279204d696.png

Please post back whatever you find. I'd like to know if you feel it's valid, if it's invalid, why it's invalid, and I'd be glad to explain how I got any particular part.

...yeah I know, reusing variables is a nasty habit but you get the idea =-P.(Nerd)