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I have always wanted to ask this question about integration by parts. Was it actually inspired by the product rule? I always use the product rule to memorize the by parts formula. Let $u$ and $v$ be 2 functions of $x$. The original by-parts function is

$$ \int u\,dv = uv - \int v\,du $$ If you start from product rule,

$$ \frac{d}{dx} uv = u\,dv\,dx + v\,du\,dx\\ \int \frac{d}{dx} uv \,dx = \int u\,dv\,dx + \int v\,du\,dx $$

Finally, by rearrangement $$ \int \frac{d}{dx} uv \,dx - \int v\,du\,dx = \int u\,dv\,dx \\ \int u\,dv\,dx = \int \frac{d}{dx} uv \,dx - \int v\,du\,dx $$

I think it was a stroke of genius by discovering this. But perhaps somebody could enlighten me.

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You're right, that's just the integration of the product rule. I was not here when it was discovered. But surely derivatives and the product rule where known before integrals. – 1015 Mar 8 '13 at 13:15
thank you @julien! was there any trace in mathematics history that this was discovered? – bryansis2010 Mar 8 '13 at 13:16
You mix thing up when you have $\frac{d}{dx}uv = udv +vdu$. It should be either $d(uv)$ on the left or $u\frac{dv}{dx}+v\frac{du}{dx}$ on the right. – Thomas Andrews Mar 8 '13 at 13:19
I really don't know. Maybe it was inspired by the summation by parts we do with series... – 1015 Mar 8 '13 at 13:21
Note, related to my comment above, $\int u\,dv\,dx$ doesn't mean anything in this context. What you mean here can be written either as:$$\int u\frac{dv}{dx}\,dx$$ or $$\int u\,dv$$ – Thomas Andrews Mar 8 '13 at 13:39
up vote 2 down vote accepted

In his book "The Calculus Gallery: Masterpieces from Newton to Lebesgue", William Dunham discusses the (very roundabout) way in which Leibnitz deduced a formula equivalent to integration by parts by considering figures.

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