100 Years to Solve an Integral (2020)

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.

The most elegant proof IMHO is the one that avoids the original problem entirely.<p>Int[csc(x) dx] = 2 Int[csc(2u) du]<p>= 2 Int[du &#x2F; (2 cos(u) sin(u))]<p>= Int[sec^2(u) du &#x2F; tan(u)]<p>= log(tan(u)) + C<p>= log(tan(x&#x2F;2)) + C<p>Then Int[sec(x)] = Int[csc(u)] = log(tan(u&#x2F;2)) + C = log(tan(pi&#x2F;4 - x&#x2F;2)) + C.<p>Of course, this was no use to Mercator, because the logarithm hadn&#x27;t been invented yet. But you aren&#x27;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.

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?

If we&#x27;re playing the map-projection-advocacy game, I&#x27;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&#x27;m not a big fan of how many equal-area &#x27;compromise&#x27; projections lie about how long the lines of latitude are.<p>[0] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mollweide_projection" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mollweide_projection</a>

&gt;[the Mercator projection] unnecessarily distorts shapes and in particular makes the Americas and Europe look much larger than they actually are. This has been linked, not without rational, to colonialism and racism.<p>The fact that on many maps Europe is much smaller that it appears should just make you all the more impressed by its achievements.

A refreshing Hacker News article after a week of repetitive political garbage. Thank you!

About how long it&#x27;d take me to solve the integral in my calculus finals.

It amuses me that doing software and hardware engineering for decades and never once thinking about trigonometric functions other than perhaps sine and cosine, and then I get interested in software defined radio and find myself running into all of the functions! That&#x27;s especially true with the discreet mathematics that SDR uses.

Oh! This was already discussed five years ago with 77 pts and 40 comments (<a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=24304311">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=24304311</a>)

This is also the inverse Gudermannian function [1]. That Wikipedia page has some nice geometrical insights.<p>[1] <a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Gudermannian_function" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Gudermannian_function</a>

I remember teaching integral of sec x to high schoolers with multiplication of sec x + tan x. I mean it is not obvious but it is not like something that would take 100 years.<p>And the author talks like logarithm was invented long after integration

Dude I am not joking but today was the day that we were introduced to indefinite integration as a formal chapter in maths at my coaching and we did secx integration.<p>Basically our sir told us to multiply &#x2F; divide by sec + tan and observe that its becoming something like integration f(x)^(-1) f&#x27;(x) * dx and if we let f(x) as t and this f&#x27;(x) * dx becomes dt Actually we can also prove the latter and I had to look at my notes because I haven&#x27;t revised them yet but its basically f(x) = t<p>so f&#x27;(x) = dt&#x2F;dx so f&#x27;(x)* dx = dt then we get<p>so integration f(x)^n * f&#x27;(x) * dx = integral t^n * dt (where t = f(x)) integral t^-1 dt so we get ln(t) and this t or f(x) was actually sec x + tan x so its ln(sec + tan) and in fact by doing some cool trigonometry we can say this as ln(tan(pi&#x2F;4 + x&#x2F;2)) + c<p>also cosec x integration is ln(tan(x&#x2F;2)) + c<p>I haven&#x27;t read the article but damn, HN, this feels way too specific for me LOL.

I&#x27;ve learned it in Austrian highschool, but then from university on nobody needed it anymore.

It feels like LLMs could be good contenders for solving symbolically integrals. After spending some time, it really feels like translating between two languages.