And so I’ve finally arrived at the end of this series, describing what one of my professors called the art of integration. I really liked that phrase, and I’ve passed that on to my own students.
I really like the integral
- Ordinary substitutions
- Trigonometric substitutions
- Trigonometric identities… lots of trigonometric identities
- The magic substitution
- Completing the square
- Eliminating unneeded parameters
- Differentiation under the integral sign (see Wikipedia for more details about this most untaught way of computing integrals)
- Partial fractions… and different ways of obtaining a partial fractions decomposition
- The substitution , converting an integral on to a contour integral over the unit circle in the complex plane.
- Converting an integral on to the limit of a contour integral over a semicircle in the complex plane.
- Residues… and different ways of computing the residue at a pole
I don’t claim to have exhausted all of the ways that this integral can be computed; please leave a comment if you think you’ve found a technique that is substantially different than those I’ve already presented.
Back when I was a student, my calculus professor said that differentiation was a science. There are rules to follow (the Chain Rule, the Product Rule, the Quotient Rule, etc.), but that any function can be differentiated through the careful application of these rules. Integration, on the other hand, is more of an art. Yes, there are some techniques that need to be known, but often great creativity is needed in order to compute an integral. Differentiation does not require much creativity, but integration does. I thought that this was a profound insight for students just learning calculus, and so I’ve been passing this insight to my own students.
There are a couple loose threads in this series that I’d like to resolve one of these days:
- I’d love to figure out a better way of showing that the above integral does not depend on without doing so much work toward computing it explicitly.
- I’d love to figure out a way of computing the integral that results after the magic substitution is performed. The denominator becomes a messy quartic polynomial, and I haven’t figured out a good way of determining the roots of this polynomial. (I avoided this complication in this series by setting , which did not ultimately affect the value of the integral.)
At the start of this series, I mentioned that this integral was original posed to me by my wife, who was trying to resolve a difference in the way that Mathematica 4 and Mathematica 8 computed it. In conclusion, I end with the Newton’s Three Laws story which was publicized in the following article that UNT publicized about my wife and me for Valentine’s Day 2015.