Integration by Parts: How Do You Choose?

In using parts to compute the integral of x^2 e^x, I chose u = x^2 and dv = e^x dx. How did I know what to choose for u and dv? First, I'll give you some rules of thumb. Then I'll illustrate these ideas by showing you examples where things go wrong.

• Don't put something in the integration column if you can't integrate it.
• Don't continue the table if things are getting worse.
• The order of priority for the derivative column is L-I-P-T-E: Logs, Inverse trig functions, Powers of x, Trig functions, Exponentials.

Suppose I choose u = 1 and dv = x^2 e^x dx. This is legal, since the two chunks multiply to give x^2 e^x. Everything starts off okay --- in fact, things look good, since the derivative of 1 is 0.

What happened? Unfortunately, to fill in the next spot in the integration column, I need to integrate x^2 e^x --- but that's the original integral! Doh!

As you can see, you should never put everything into the integration column. In general, don't put something in the integration column if you can't integrate it.

Suppose I choose u = e^x and dv = x^2 dx. This is legal, since the two chunks multiply to give x^2 e^x.

The derivative of e^x is e^x, the derivative of e^x is e^x, the derivative of e^x is e^x ... Hmm! Since this isn't going to stop by itself, I'll move to the integration column.

I compute the first three integrals, and ... Hey! The powers of x just keep getting bigger.

The table isn't going to stop by itself. But if I try to stop by integrating across a row, I have to integrate a power of x times e^x, which is what I had to begin with.

When you do math, you should always step back and look at the big picture now and then. Are you making progress? Are you going nowhere? In this case, the bigger powers of x are a sign that I made a bad choice for "u" and "dv" at the start. Don't continue the table if things are getting worse.

I've shown you how a couple of bad choices work out. Try some other choices for "u" and "dv". See for yourself why different things go wrong.

My original (correct) choice was u = x^2 and dv = e^x dx. I was following the L-I-P-T-E rule. x^2 is a Power and e^x is an Exponential. Since "P" comes before "E" in L-I-P-T-E, Powers have priority over Exponentials in the derivative column. So the "x^2" goes in the derivative column, and that leaves the "e^x" to go in the integral column.

• Introduction: Integration by parts
• How do you set up an integration by parts table?
• The integral of x^2 e^x
  • How do you choose what to put in the table?
• The integral of ln x
• The integral of e^(2x) cos (3x)
• A recursion formula for the integral of (sin x)^n
• Computing a definite integral using integration by parts

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Last updated: June 13, 2005

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