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:

• Integration by Parts (YouTube), a wonderful demonstration that clarifies the technique by way of a couple easy examples
• Calculus: Techniques of Integration — Integration by Parts
• Integration By Parts by Duane Kouba
• Paul’s Online Math Notes: Calculus II – Integration by Parts
• Wikipedia: Integration by parts

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:

• Wikipedia: Brook Taylor
• Taylor biography (University of St Andrews, Scotland)
• Brook Taylor (a brief overview by Erich Friedman, Stetson University)

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:

• Wikipedia: Taylor series
• YouTube: Taylor and Maclaurin Series, Example 1
• Finding Taylor Series

And to bring Taylor series and integration of parts together:

• Wikipedia: Taylor’s theorem
• Neat Tricks Chapter 2 — Taylor Series or Integration by parts the wrong way … compare with this Derivation of the Taylor Series
  
• Adrian Down (January 24, 2006) gives us Taylor Series [PDF]
• Padraic Bartlett provides us with “Integration by Parts / Taylor Series / L’Hôpital’s Rule [PDF; homework discussion]
• Integration by parts and infinite series