Page:Calculus Made Easy.pdf/246



Dodges. A great part of the labour of integrating things consists in licking them into some shape that can be integrated. The books–and by this is meant the serious books–on the Integral Calculus are full of plans and methods and dodges and artifices for this kind of work. The following are a few of them.

Integration by Parts. This name is given to a dodge, the formula for which is

$\int udx=ux-\int xdu+C$.

It is useful in some cases that you can’t tackle directly, for it shows that if in any case $$\int xdu$$ can be found, then $$\int udx$$ can also be found. The formula can be deduced as follows. From p. 38, we have, $d(ux)=udx+xdu$, which may be written $u(dx)=d(ux)-xdu$, which by direct integration gives the above expression.