A technique for solving some indefinite integrals. Based on nothing more than the derivative law for products:
(1)  (u*v)' = u'*v + u*v',
it can solve a great many integrations.

To derive the formula, rewrite (1) as

(2)  u'*v = (u*v)' - u*v'.
Integrating both sides, we see that
(3)  integral u'*v dx = u*v - integral u*v' dx
where we hope u*v' is easier to integrate than u'*v.

A more concise way of writing (3) is to use the formal notation du=u'dx, dv=v'dx, and use that to re-write (3) as

(4)  integral v du = u*v - integral u dv