If you draw a graph it’s nice to get some intuition for why: the area under the inverse is the area to the left of the curve up to a given point. This is equal to the area of the “rectangle” minus the are under the original curve :D
Interesting to find another vegan mathematician on Lemmy!
😉
it’s beautiful how such a random thing could have such a simple, yet hard to stumble upon explanation…
deleted by creator
glad this is helpful for someone :)
btw, I just spotted a little error in integration of arctan(x), it’s supposed to be
x*arctan(x)-tan'(cos(arctan(x)))instead ofx*arctan(x)+tan'(cos(arctan(x))), fixed nowdeleted by creator
I think the derivation you wrote is wrong but the outcome is correct. Looks like a typo. Wouldn’t the integration by parts step give us u\mu(u) - \int \mu(u)du ?
Also I believe that this is possibly the building blocks of the Laplace Transform. It looks really similar at least.
oops, yeah, that’s a typo, thanks for pointing out, i fixed it now
