Integration by parts says
The first question students ask is What do I make ‘u’ and what do I make ‘dv’? I used to tell my students to set ‘u’ equal to the part you’d rather differentiate and ‘dv’ equal to the part you’d rather integrate. That’s not bad advice, but it begs the question “How do I know what I want to differentiate and what I want to integrate?” Until you have some experience and intuition, that’s hard to answer.
Here’s a good rule of thumb: set ‘u’ to the first term you see on this list:
- inverse trig function
- algebraic function
- trig function
This rule doesn’t cover everything — no rule can — but it works remarkably well. I don’t remember just where I found this; I believe it was in an article somewhere. I’m fairly certain I’ve never seen it in a calculus textbook.
Update: I found the reference for the rule above. “A Technique for Integration by Parts” by Herbert E. Kasube. American Mathematical Monthly, March 1983, page 210.