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I was recently refreshing my memory of integration by parts via some personal reading when I thought, there must be a better way. Integration by parts (while creative in that it integrates the entire product rule) feels very arbitrary to what it's attempting to calculate (at least concerning the integral of a product of two functions).

So, I ask, are there any alternatives to integration by parts? Perhaps, any less messy methods? Don't take this the wrong way: integration by parts is brilliant. But I still feel that there has to be a simpler, much more intuitive way to calculate the integral of the product of two functions.

If there aren't any defined alternatives, is there any work being done by any mathematicians (publicly) on a better way to complete said task?

I personally spent a few hours trying to make some connections that might lead me finding an alternative method, I didn't find an alternative method.

Thanks!