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.

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January 3, 2020

Number of bijections with no fixed points

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 \(!n\).

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