# The Unapologetic Mathematician

## 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 $\left[fg\right](x)=f(x)g(x)$ is given by the “Leibniz rule”: $\left[fg\right]'(x)=f'(x)g(x)+f(x)g'(x)$. Now we take the antiderivative of both sides:

$\displaystyle f(x)g(x)=\int\left[fg\right]'(x)dx=\int f'(x)g(x)dx+\int f(x)g'(x)dx$

Adding specific limits of integration and rearranging a bit we find the usual formula for integration by parts:

$\displaystyle \int\limits_a^bf(x)g'(x)dx=f(b)g(b)-f(a)g(a)-\int\limits_a^bf'(x)g(x)dx$

So if we can recognize our integrand as the product of a function $f(x)$ that’s easy to differentiate and a function $g'(x)$ 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 $f$ and $g$ at the boundary points $a$ and $b$.

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.

February 25, 2008 - Posted by | Analysis, Calculus

1. I’ve had almost more integration by parts than I can stand during the last few weeks, with all the calculus tutoring I do. (Perhaps I should send my students to this site.)

This is one of my favorite techniques, because I love how it can help solve some seemingly impossible integration problems, especially some particularly horrible ones that you can’t imagine could possibly have a “nice” integral.

Comment by musesusan | February 25, 2008 | Reply

2. Integration by parts is one of my least favourite techniques. Unless I specifically need some reduction formula, I don’t use it if I can avoid it.

(IME, any integrand which would succumb to integration by parts succumbs more easily to a Risch-Norman attack.)

Comment by Pseudonym | February 26, 2008 | Reply

3. Pseudonym: I actually don’t compute many antiderivatives at all, myself. You’re right that there are often other methods. On the other hand, I know I’m going to use it at certain points in the future.

Besides, it’s a great illustration of the use of FToC as a mirror.

Comment by John Armstrong | February 26, 2008 | Reply

4. [...] of Variables Just like we did for integration by parts we’re going to use the FToC as a mirror, but this time we’ll reflect the chain [...]

Pingback by Change of Variables « The Unapologetic Mathematician | February 27, 2008 | Reply

5. [...] where else have we seen derivatives as factors in integrands? Right! integration by parts! Here our formula says [...]

Pingback by The Riemann-Stieltjes Integral IV « The Unapologetic Mathematician | March 6, 2008 | Reply

6. [...] where again we throw away negligible terms. Now we can handle the first term here using integration by parts: [...]

Pingback by The Higgs Mechanism part 1: Lagrangians « The Unapologetic Mathematician | July 16, 2012 | Reply