I’ve been looking forward to sharing two calculus ideas from Patrick Honner with my son for the last week. We were, unfortunately, a little rushed when we sat down and there are a couple of mistakes in the videos below. Even though things didn’t go perfectly, I really enjoyed talking through these ideas.

Here’s the first idea – a twist on integration by parts that Honner learned from the British mathematician Tim Gowers:

Here’s the second idea – a fun surprise when a student made a creative substitution in a integration problem:

So, I stared the project by talking about how to integrate arctangent without using integration by parts:

In the last video we found a possibly surprising connection between arctan(x) and ln(x). Here I introduced the integral from the 2nd Patrick Honner tweet above and showed my son how you solve that integral using partial fractions. The point here wasn’t so much the integral, but rather to show that ln(x) showed up in an integral similar to the one we looked at in the first part of the project:

How I showed the technique that Honner’s student used (though I goofed up the substitution, unfortunately, using u = ix rather than x = iu. By dumb luck, that mistake doesn’t completely derail the problem because it only introduces an incorrect minus sign):

Now that we’ve found two connections between arctan(x) and ln(x), we went to Mathematica to see if the two anti-derivatives were really the same. It turns out the are (!) and we got an even bigger surprise when we found that Mathematica uses the same technique that Patrick Honner’s student used 🙂

Also, in this video I find a new way to introduce a minus sign by reversing the endpoints of an integral . . . . .