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.
Since we’ve established the connection between integration and antidifferentiation, we’ll be concerned mostly with antiderivatives more directly than derivatives. So, it’s useful to have some simple notation for antiderivatives.
That’s pretty much what the “indefinite integral” amounts to. It looks like an integral, and it does (what the FToC tells us is) all the hard work of integration, but it stops short of actually calculating an integral. Given a function , we write an antiderivative as . Note that we aren’t saying which antiderivative we mean, and for the purposes of the FToC (part 2), we don’t need to be. It’s customary, though, to write the result generically by adding a to the end of it.
We know, for example, that
Then we turn this around to write
and so on.
We can also go back and rewrite the two rules of integration we found before:
Notice here that we don’t need to add the , since each side consists of indefinite integrals. We can hide these “constants of integration” on both sides. They only need to show up once we fully evaluate an indefinite integral.
Last Monday I noticed an XKCD comic and then later deconstructed it. The upshot is that I didn’t like it, but many XKCD fans turned around to tell me that I was either stupid or crazy to question Randall’s artistic vision.