Q: Who invented integration by parts?

A: Brook Taylor.

Online a friend inquired, “Do you know who invented integration by parts?” Off the top of my head I did not. Did I ever? I was not sure. Did I have suspicions? Sort of.

I. Integration By Parts

See the following links for an overview:

In short: you have a relatively nasty looking function of a variable ‘x’ that you don’t know how to integrate. However, if you can rewrite it into a composition of nicer looking functions by way of substitution, say functions ‘u’ and ‘v’, so that one is easy to integrate and what would be hard to integrate is at least easy to differentiate, you can probably land at a solution by dealing with ‘du’ and ‘dv’ rather than ‘dx’.

Its form reminds one of the product rule for differentiating, as it should, as it can be derived from it.

It’s a strategy for dealing with non-obvious problems but it does not guarantee a solution. It can be generalized: you can have recursive integration by parts, and you can also generalize to higher dimensions, to functions of several variables. And it’s used frequently for trigonometric functions.

But what the webpages above won’t tell you is ‘who invented’ (or ‘discovered’) it. On the one hand … it doesn’t need much in the way of discovering, as it follows from the product rule, which is attributed to Leibniz and occasionally to Isaac Barrow. As Will Garner writes on his page on the topic, “When I was first introduced to the formula for integration by parts, I was never really told where it came from,” but he continues, “The origins, however, are useful in not only understanding but also remembering the formula.” Garner then connects the product rule for derivatives to integration by parts; it’s the completion of the parallelism that is found in the power and chain rules for each of derivatives and integrals. This helps to — pardon the pun — derive integration by parts and contextualize it, but does not historicize it.

II. Brook Taylor

See now:

Both of the last two links give Taylor credit for integration by parts, as does the Encycopédia of Mathematics by James Tanton: “Taylor also invented the technique of integration by parts” (494). His entry followed ‘tautology’. This is irrelevant.

Taylor’s name was already familiar and I felt an inkling of recognition when thinking about integration by parts because we know Taylor from the ‘Taylor series‘ that are named after him (but with which he was not the first to work: “Despite the attachment of his name to the technique, Taylor was not the first to develop a theory of infinite function expansions” (ibid)).

See also:

And to bring Taylor series and integration of parts together:

About Steve

47 and counting.
This entry was posted in Q&A. Bookmark the permalink.

2 Responses to Q: Who invented integration by parts?

  1. surfer says:

    I invented them. (Re-invented them, that is.) Was super, super into AP calc in the early 80s (working 100% of the homework before lectures, getting extra books). I actually came up with it a few weeks before it was introduced because my teacher had listed some complicated integral that she said we could not figure out, but it was clearly a product. Realized that I could sort of make one thing go away and one thing stay and then did about 6 successive integration by parts to figure it out. Then generalized the idea. We hadn’t gotten to IBP yet in the book (few weeks later) but I just came up with the concept. When you work so many problems, sometimes you can (re)invent a trick or two before taught it.

    Reported back to my teacher and everyone in the class was stunned. Didn’t really invent anything else, though. Well, one time I solved a shell volume problem with washers and the teacher was mad at me, because I covered her whole lecture for the day in the course of showing a HW problem. But that was from always being a day or two ahead, not from invention.

    • ninjaballerina says:

      What a fascinating idea- although I’m afraid it’s impossible that you “reinvented” integration by parts. I understand and sympathize with your confusion-the term “reinvented” does sound as though it means “to invent again”, but to invent something for a second time is impossible. Once something has been invented, or created for the first time, it cannot be invented, or created for the first time, again. That is why the correct use of the term “reinvention” refers to the act of changing something so much that it appears to be entirely new.
      In fact, the question of who invented integration by parts should not even have been posed in the first place. The question should have been one of who DISCOVERED integration by parts, for all math is discovery, and not invention.

Leave a Reply

Your email address will not be published. Required fields are marked *