What to make "u" in integration by parts

Integration by parts says

integration by parts

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:

  1. logarithm
  2. inverse trig function
  3. algebraic function
  4. trig function
  5. 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.

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9 comments on “What to make "u" in integration by parts
  1. Ryan says:

    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.

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

  3. Glenn Decker says:

    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

  4. John says:

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

  5. Chris says:

    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

  6. I. J. Kennedy says:

    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!

  7. Jake says:

    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.

  8. Jake says:

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