Scalar Integration by Parts

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

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

Integrate both sides.

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

and rearrange:

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

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 $g(x)$ :

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

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