Integration by Parts

Integration by parts is an integration technique which is based on the Product Rule for derivatives. The Product Rule says:

Integrate both sides, then rearrange the equation:

The last line gives the Integration by Parts formula:

In order to make computations easier --- much easier! --- I'll set up parts problems using tables, as opposed to applying the formula above directly.

• 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

I first saw the tabular approach to parts in Horowitz's article; he refers to the earlier article by Murty. I've adapted the "L-I-P-T-E" rule from an article I read a while ago --- but I can't find the reference! I'd be grateful if someone who remembers the article would let me know.

I read Horowitz's article a few summers ago. At the time, I was tracking a group of students through the first 4 terms of calculus at Case Western Reserve University, and I had shown them parts the traditional ("u - dv") way during the spring term. When we started multivariable calculus in the fall, I began by reviewing integration techniques. I told them that I had found a new way of doing parts which I thought was pretty neat, and showed them a couple of problems using the tabular approach.

Students often resist learning old things in a new ways, but this time I got a room full of astounded faces! Finally, someone said: "It's so easy!" With that kind of reaction from people who had seen both methods --- as well as the obvious pleasure people took in the tabular approach --- I knew I had a winner.

• David Horowitz, Tabular Integration by Parts, College Mathematics Journal, 21(4)(1990), 307--311.
• V. N. Murty, Integration by Parts, Two-Year College Mathematics Journal, 11(1980), 90--94.

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Last updated: June 13, 2005

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