Integration by Parts
Now we can use the FToC as a mirror to work out other methods of finding antiderivatives. The linear properties of differentiation were straightforward to reflect into the linear properties of integration. This time we’ll reflect the product rule through the FToC to get a method called “integration by parts”.
The product rule tells us that the derivative of the product of two functions is given by the “Leibniz rule”: . Now we take the antiderivative of both sides:
Adding specific limits of integration and rearranging a bit we find the usual formula for integration by parts:
So if we can recognize our integrand as the product of a function that’s easy to differentiate and a function that’s easy to integrate, then we might be able to simplify things, though we have to be careful about the new terms that crop up from evaluating and at the boundary points and .
As a side note, physicists love to use this technique (and more general analogues) by waving their hands hard enough to push the boundaries far enough away that they can be ignored. There are some — like my departmental colleague Frank Tipler — who think this is the source of most problems modern physics seems to have. Myself, I take no position on the matter. I’ve upset enough people for this month already.