Pardon the fact that I may be butchering the proper name of the general rule for the below indefinite integration in the title (and please feel free to edit the heading):
$$\int((f(x))(g(x)))\>dx\>=\>g(x)*\int(f(x)\>dx)-\int[\int((f(x)\>dx)\>*(\frac{d}{dx}g(x))]\>dx$$
I've been struggling conceptualizing the rule and would love if someone could logically and pre-algebraically prove it.
Imagine the product rule for derivatives $$(uv)' = u' v + u v'.$$ If we integrate and rearrange we obtain the rule for integration by parts. $$ \int u' v \, dx = uv - \int uv' \, dx. $$