I have a much younger sister who is currently taking calculus as a high school junior. Given that I have a little more mathematical training than anyone else in my family, she sometimes asks me for help with her homework. This is almost invariably a humbling experience.
Last week, she was having trouble with the following indefinite integral:
\(\displaystyle
\int \frac{{\rm csch}(\ln(t))\coth(\ln(t))}{t}\,dt
\)
If you are unfamiliar with calculus, don’t worry about it—I don’t plan to go that deep into the topic today. The point of today’s exercise is to demonstrate how helpful it can be to keep in mind the old axiom, “Keep it simple, stupid!”
Here was my first thought: “Hyperbolic trig functions? Hrm… let’s unpack those.” The functions in the numerator are hyperbolic trigonometric functions. These functions are actually defined in terms of exponential functions (i.e. \(e^x\)), but they behave in a manner similar to trigonometric functions with respect the derivatives, so there is some logic in calling them “trigonometric” functions. My first impulse was to unpack the definitions, and rewrite the numerator in terms of exponential functions.
This wasn’t actually a terrible idea. The argument of the two hyperbolic trig functions are natural logarithms, and (for instance) the hyperbolic cosecant of a natural logarithm simplifies down to a rational function (i.e. a fraction with polynomials in the numerator and denominator). In some ways, rational functions are easier to work with, so my instincts weren’t horrible.
After replacing the hyperbolic trig functions with rational functions, I was able to do some simplifications, and reduced the integrand to a rational function. This was not a nice function, and there really wasn’t an easy way of working with it, but I had already committed myself to one approach, and I was damned if I wasn’t going to keep going.
Rational functions aren’t too bad to integrate if the denominator is “nice” in some fashion. The usual technique for working with such functions is to decompose the fraction by way of partial fractions. It has been a while since I took calculus, so I had to look up the technique online, but after a few minutes, I had it down.
Of course, it turned out that there were several possible decompositions of the function that I was working with, so I made many false starts before gaining ground. However, after banging my head against the wall for a good 45 minutes, I finally had a decomposition that was useful. Success!
Finishing the problem required a couple of quick substitutions and some simplification. I ended up with a neat little rational function, and all it took was three hours of false starts and complex computations. Now, being a clever student, I looked up the answer in the solutions manual, and discovered that my answer looked nothing like the answer given (which was \(-{\rm csch}(\ln(t))\)).
Frustration! Back to the drawing board. But first, a quick check: after unpacking the “correct” solution, I was able to confirm that my answer matched. Whew! Crisis averted.
It was at this moment, after completely filling my whiteboard and struggling with the problem for hours, that one of my colleagues walked by my office, and made the following comment: “Why didn’t you just do a substitution in the first step?” After five minutes of work and four lines of computation, I had the correct answer. All that sturm und drang, and the solution was simple, elegant, and obvious. Why hadn’t I seen that?
At the very beginning of the process, I managed to convince myself that the problem was far more complicated than it actually was, and let that conviction lead me down a garden path. As always, the correct route was to keep it simple. D’oh!