It 'Cannot' Be Done (Integrals)

Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

[BriTheMathGuy]

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[Jj-gisg]

Me an intellectual: "Oh its obviously e^x²/2x "

[tomasstana]

A long time ago, with my friend at high school we once felt bored and unchallenged with the integrals we were computing for homework, so we decided to pick one integral we already finished and make it more difficult. The one we picked was x*e^x^2 and we dropped the x.

Two hours later we gave up and admitted defeat ...

[ClumpypooCP]

I like these funky integrals... do more of them

[Dongerd]

I just saw that the integral had the word “sex” and clicked

[MrCigarro]

Thank you. For us statisticians this is a very important function.

[EebstertheGreat]

A more precise way of stating this idea is that exp(x²) has no _elementary_ antiderivative. Roughly speaking, elementary functions on the complex numbers consist of the rational functions and finitely many extensions by exponential and logarithmic functions. On the real line, restrictions of these are also usually included, such as the trigonometric functions (as well as their inverses). That is, a real-valued function of real numbers is elementary iff it is a composition of finitely many functions in the set {+, ×, exp, log, sin, arcsin} and constant and projection functions of the real numbers.

An even more precise statement is that the function f satisfying f(x) = exp(x²) has no primitive in any elementary extension to the differential field of rational functions (i.e. not in any differential field which can be obtained by a finite chain of logarithmic, exponential, or algebraic extensions starting with the rational functions), which can be checked by applying Liouville's Theorem and solving the resulting differential equation.

[IngTomT]

I would keep the factorials, I think it looks nicer in the form
{ Sum(n=0, oo): x^(2n+1) / (2n+1)n! } + c

[hardchemist]

That was great and funny as hell at the same time lol

[EebstertheGreat]

Also, you mentioned the imaginary error function erfi. The exact definition is erfi(z) = -i erf(iz) = 2/√π ∫ exp(t²) dt, where the integral is taken from 0 to z. In other words, it is a scaled version of the antiderivative you supplied when c = 0, continued to the entire complex plane. (I find the 2 in the numerator irritating. I don't really know why it's there. I guess it's supposed to be easier to construct confidence intervals with erf this way.)

[justinberdell]

Fascinating! I never saw it explained like that. And in 3 minutes no less! Lol I have however used the error function a lot. I'm an electrical engineer and in undergrad I took a class in thin film semiconductor fabrication and the error function is used to calculate dopant or impurity concentration in constant source diffusion. Really cool stuff but I went into ai and robotics so I never use it now

[henrytang]

I like how they had to invent a completely new function to describe this integral.

[YoutubeUser-ylys]

Y’all didn’t pay attention how much he would struggle to write in reverse in front of him so we see directly.

[Zeebzz]

I really thought it was (e^x)^2 and I was like why did you make it so difficult, then realized it was e^(x^2)

[ezequielgerstelbodoha]

I just found so anti-intuitive that the integral of xe^(x^2) is fairly simple to solve, that e^x is one of the easiest ones, but e^(x^2) doesn't have an elementary answer. I was breaking my head around a couple substitutions when I searched for this

[lht]

Euler: It is trivial that 2pi Exp[z^2+y^2] dzdy = 1/2 Exp[r^2] dr^2 dtheta in polar coordinate, 2pi cancels dtheta, so the integral solves to Exp[x^2]/2x + C

[alexplastow]

Thanks for being casually excited about mathematics

[UdayadityaSankarDas]

I encountered this function while doing differential equations and it made me lose a question in exam. I've been both fascinated and scared of this one since then.

[jaskaransingh]

Well if u integrate this u will get the integral of erfi. And I integrated to be as so that the integral is equal to (sqrt(pi) erfi(x))/2 +c 😊

[mysticdragonex]

Better way will be to use the Gamma function I think