Dear Perfect H:

I like your method of integration by parts a lot. I haven't tested it yet, so I am still left with one question: is this a complete replacement for u substitution? If so I am very pleased!

Another thing I'm wondering is, are there ever cases in which we can't do the first step, i.e. in which we can't turn one of the factors into a (solvable) derivative? If so, what then?