some indefinite integral
s. 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