January 4, 2020

Integrating a rational function with a convenient power term

Once upon a time in a Calculus class, I was trying to evaluate the following integral: $\int \frac{x^2 + 8}{x^3 + 9x} \, \mathrm{d}x.$ The obvious way was to do a partial fractions decomposition, but I was feeling lazy and decided to use WolframAlpha to compute it and check the step-by-step solution to make sure I had the correct idea. Instead of using partial fractions, WolframAlpha gave an interesting alternative solution.

To test my new blog with MathJax integration, I'll post an old exercise proving that the number of bijections $f:A\to A$ with no fixed points equals $!n$.