I had a post the other day about the product rule in differential calculus, and how to extend it to find the derivative of a reciprocal and a quotient. The product rule also does heavy duty in the opposite direction, in the method called integration by parts. Strang has a wonderful diagram of that (shown above), which I've reproduced here using matplotlib.
The product rule is:
Integrating both sides and isolating u v' we get:
where for the definite integral we have to remember to evaluate u times v at both of the limits of integration, subtracting the value at the lower limit from the upper one.
Can you see each of these four terms as areas in the diagram? It's very pretty.