Just in case a picture helps...
... is lazy integration by parts, doing without u and v.
... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x.
The formula, of course, (one version, anyway) is
But the important thing is to see how it depends entirely on working backwards through the product rule for differentiation.
Also: Integration by parts - Wikipedia, the free encyclopedia
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!