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:

- logarithm
- inverse trig function
- algebraic function
- trig function
- exponential

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.

That’s really interesting. The trick I always use is, let dv be the function that has the cleanest antiderivative such that the order does not increase (unless using the “invisible dv”)

I love this part of Calculus 2. A nice taxonomy of integration tricks, and integration by parts has its own corner cases such as using “I” and the “invisible dv” where dv = dx.

My students taught me this 8ish years ago. They remembered it with the mnemonic L.I.A.T.E.

Stumbled upon your web page, looks like some useful tips to be found here. But I must say you’re guilty of misappropriating the term “begging the question” which is a formal logical fallacy whereby the conclusion of an argument is assumed in the premises. You meant “raises the question”. Just my grammar nazi contribution of the day. GD

Glenn: Touché. I can be a grammar stickler too so I appreciate the tip.

Dear John

Just in the interest of math education trivia (perhaps a better word is in order):

I’m a Singaporean (you know… a dude, from Singapore.) We sit for a modified version of the A-levels (what they take over in England) and at least here teaching the LIATE schema is common. I am not too sure about England.

Best

Chris

I’m also one of those grammar Nazis who cringes when I see a misuse of

beg the question, and I did pause when I came across it in your post. But upon further reflection, I decided it was a proper use after all—begging the question means answering the question with another, essentially equivalent, question. Let’s give him a pass, Nazis!Solved exercises of

Immediate integrals

Integration by substitution

Integration by parts

I’ve only recently figured this out, but at least for polynomial u or v, it’s best to set u to whichever function you want to reduce in power in order to simplify the integrand.

So, for example, to integrate x^2 * e^x with respect to x, set u = x^2 and integrate by parts twice.

Even though it’s only a special case, I’ve found this generates more insight with my students than the line about which function you’d rather differentiate, etc., which is what I was told as a kid and never really found helpful.

Also: This is an excellent rule. I’m definitely going to be using it in my tutoring.