100 Years to Solve an Integral (2020)
20 comments
·April 20, 2025ziofill
dataflow
The impulse train is another well-known one, though I suppose someone will chime in here to rebut that it's arguably not a function.
cl3misch
Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?
seanhunter
Trig is full of functions that fall into disuse and are forgotten.
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
Suppafly
>Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function
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.
scotty79
I'm sure you used inverse of a cosine multiple times. Didactic math today is just not bothering to give it a name. Probably because people think that sin, cos and tan is enough. Even ctg which is just inverse of tan is often skipped.
seanhunter
I know what you mean, but as a sibling pointed out for everyone else's benefit, parent is using the word inverse where they mean reciprocal.
The inverse of cosine is arccosine (sometimes written acos or cos^{-1}). Secant is the reciprocal of cos ie sec x = 1/cos(x)).
Likewise cotan is the reciprocal of tan (1/tan). The inverse of tan is atan/arctan/tan^{-1}.
This is confusing for a lot of people because if you write x^{-1} that means 1/x. If you write f^{-1} and f is a function, then _generally_ it means the inverse of f. In the case of trig functions this is doubly confusing because people write sin^2 theta meaning (sin theta)^2 but sin^-1 theta means arcsin theta.
That's why in my maths studies they started by teaching you to do the inverse with a -1 so when you see it you don't get confused but changed to preferring arcsin etc as this is unambiguous and if you learn to write this way you won't confuse others.
asplake
The secant is the reciprocal of a cosine – the hypotenuse over the adjacent
anyfoo
That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.
LegionMammal978
If we're playing the map-projection-advocacy game, I'd say the Mollweide projection is underrated among equal-area maps [0]. (For local maps, use whatever you want, appropriately centered.) Sure, it distorts shapes away from the central meridian, but locally it only adds a simple horizontal skew. I'm not a big fan of how many equal-area 'compromise' projections lie about how long the lines of latitude are.
billab995
About how long it'd take me to solve the integral in my calculus finals.
redbell
Oh! This was already discussed five years ago with 77 pts and 40 comments (https://news.ycombinator.com/item?id=24304311)
scythe
The most elegant proof IMHO is the one that avoids the original problem entirely.
Int[csc(x) dx] = 2 Int[csc(2u) du]
= 2 Int[du / (2 cos(u) sin(u))]
= Int[sec^2(u) du / tan(u)]
= log(tan(u)) + C
= log(tan(x/2)) + C
Then Int[sec(x)] = Int[csc(u)] = log(tan(u/2)) + C = log(tan(pi/4 - x/2)) + C.
Of course, this was no use to Mercator, because the logarithm hadn't been invented yet. But you aren't just pulling a magic factor out of nowhere. There is definitely a bit of cleverness in rearranging the fraction — you have to be used to trying to find instances of the power rule when dealing with integrals of fractions.
qbane
This was the one I was taught in my high school. It has some cleverness (e.g., some trig. transformations) but looks less like coming out of nowhere than the original.
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caminanteblanco
[flagged]
whatever1
It feels like LLMs could be good contenders for solving symbolically integrals. After spending some time, it really feels like translating between two languages.
331c8c71
Wolfram engine was taking integrals just fine way before LLMs were even a thing.
anthk
And Lisps too, fitting a sector from a disk:
https://justine.lol/sectorlisp2/
And probably a small forth too, with a dictionary defining every math word, something not so different to Lisp.
LLM's? 4GB of RAM? Your grampa's 486 with 16MB of RAM can do calculus too.
2b3a51
Derive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample.
You can do a lot of numerical maths just with a noddy spreadsheet of course.
I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.