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

Integrate both sides.

and rearrange:

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

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