Mathematics has a lot of integrals in it.

Some of them are kinda hard to compute, like $\int (\sin x + \cos x) e^x\,dx$, where you have to use integration by parts and rearrange or something. Some of them are really hard to compute, like $\int \sqrt{\tan x} \,dx$, where you have to perform many unintuitive substitutions in sequence to obtain an unenlightening and kind of disgusting answer.

But some of them are impossible, like $\int \sin \sin x \,dx$. There is a mathematical proof of this fact: Liouville’s theorem. Now, I can understand hard things being hard, for necessity or even for the sake of it, but this is just crossing the line. Totally unacceptable.

I rate integrals 0 out of 5 stars. Would not recommend.

This is the debut post on my ‘blog, which for I have called ‘The Clearest of Blue Skies’. This is a reference to one of my favourite games, Tetris Attack, but it’s also a nod to one of my goals for this ‘blog, which is to exposit mathematics as clearly and concisely as possible.

My very near-sighted motivation for starting this ‘blog is to work out some things that I haven’t seen consolidated elsewhere, and I have more than a couple posts on such things already in the works. But I also have a number of old bits I’ve written about math that I’ve been meaning to peruse, now that I’m better at understand what I’m talking about, so I’d like to format that as well.

I hope not to be inaccessible in my rants, but I realize that a lot of the stuff I’d like to talk about is pretty obscure even for many mathematics undergraduates. So I’ll do my best but I make no promises.