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Scalar Integration by Parts

We have already done almost all of the work here. Start with the product rule for the differential:

$\displaystyle d(fg) = f\ dg + g\ df $

Integrate both sides.

$\displaystyle \int_a^b d(fg) = fg\bigg\vert _a^b = \int_a^b f\ dg + \int_a^b g\ df $

and rearrange:

$\displaystyle \int_a^b f\ dg = fg\bigg\vert _a^b - \int_a^b g\ df $

This is one way of writing integration by parts, but we aren't usually given both ``$ df$ '' and/or ``$ dg$ ''. Note well that we can express $ df$ and $ dg$ in terms of the chain rule, though, which is exactly what we will usually be doing to express the integral of known functions $ f(x)$ and $ g(x)$ :

$\displaystyle \int_a^b f\ \ddx{g} dx = fg\bigg\vert _a^b - \int_a^b g\ \ddx{f} dx $

Integration by parts is an enormously valuable tool in scalar/one dimensional integral calculus. It is just as important in multivariate integral calculus!

Robert G. Brown 2017-07-11