If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that regard?

Makes sense, given that the definition of e goes hand in hand with its property of e^x being its own integral and derivative.

For example "versine"

versin theta = 1-cos theta.

There is also "haversine" which is (1-cos theta)/2. Which is used in navigation apparently https://en.wikipedia.org/wiki/Versine

I think we used it in geometry in US high school, but only to complete an assignment or two to show we could use trig functions correctly. I had to relearn how all of them worked to help my kid with homework, it's mostly look at the angles and sides you have available and pick which trig function is necessary to figure out which one you're solving for. I'm sure there are real life uses for trig functions, and I hate to be one of those "when are we ever going to use this" types, but I've never used any of them outside of math classes.

and in fact our sir himself told us that he would've also let us do this if we were in normal batches (we are in a slightly higher batch, but most students are still normal and it was easy to digest to be honest except when I was writing this previous comment, I actually found that our sir had complicated the step of f'(x) = df(x)/dx by letting us assume f(x) as t and so on..,maybe it makes it easier to understand considering f(x) to be its own variable like t instead, but that actually confused me a little bit when I was writing the previous comment) , still nothing too hard.

I actually want to ask here because I was too afraid to ask this to sir, but is there a way, a surefire way to solve any integral , like can computers solve any integral?