Why can't you integrate e^(x^2)? >Undergrad calc student

>>8462064

Find me a function such that when I take its derivative it'll give me e^(x^2).

You can always write it as a Maclaurin series though, and integrate that.

t. undergrad

>>8462064
You can!

>>8462067
How would one calculate the area under the curve?
>>8462068
Why am I in a numerical methods class then?

>>8462072
>Why am I in a numerical methods class then?
To learn how to integrate it?

>Why can't you integrate e^(x^2)?
it's not exactly trivial
https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)

>>8462079
we are using Eulers formula to approximate a solution....

>>8462095
>we are using Eulers formula to approximate a solution....

Now take the limit as the step goes to 0

>>8462072

>how would one calculate the area under the curve

See my second point.

Or just use a computer like everyone in the real world does.

>>8462064
What are you talking about? You can integrate any Lebesgue-integrable function, and [math]x \mapsto \exp(x^2)[/math] is trivially so. The solution is of course the function [math]f \colon \mathbb{R} \to \mathbb{R}[/math], defined pointwise by [eqn]f(x) := \int_{0}^{x} \exp(t^2)\,dt.[/eqn]

Perhaps your question is "why can't I express the values of [math]f[/math] in terms of 'standard functions'", to which I say, "define what a standard function is." Then we get into arguments about whether an infinite power series representation is "standard", whether [math]\mathrm{Li}[/math] or hypergeometric functions or even Bessel functions are standard, etc. etc. This becomes not a question of mathematics anymore, at least not in my opinion; it's a question of standards and definitions.

That said, there are still many interesting discussions about the notion of a "closed-form expression", as it is often called, see http://www-math.mit.edu/~tchow/closedform.pdf . However, the mathematics thrown around to make such ideas concrete are often beyond the scope/interest of the "undergrad calc student", as you call yourself.

>>8462455
Don't be a cunt. Just swallow your pride and say you don't know. We're all VERY impressed you've taken Lebesgue theory by the way.

OP, the reason is very complicated and not easy to explain... especially on 4chan. It'll be a tough read, but you can see this reference if you are really interested http://projecteuclid.org/euclid.pjm/1102991609

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>>8462500
>We

>>8462455
> any Lebesgue-integrable function
ah but anon, as you surely are aware, 'Lebesgue integrable' means that the function is contained in the space [math] L^1(\mathbb{R})[/math], which the function in question is not. Presumably you meant to say "locally integrable" :^)

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>>8462064
>Why can't you integrate e^(x^2)?
Because you're retarded I guess

>>8462455
>any integrable function is integrable

>>8462859

I know it sounds trivial but that's pretty much the point. "Is this function integrable?" is a totally different question from "does the antiderivative of this function have a closed-form expression."

>>8462870
You can integrate it because it's continuous.
You're saying you can integrate it because you can integrate it

>>8462948
Well yes, integrability is not concerned by antiderivatives.

>>8462455
Thanks, I thought it strange that we were simply told "can't be done" I though there must have been some theory we didn't know.
>>8462500
I'm interested, thanks for the reference

How do I use Bolzano method to solve pre-defined polynominal function, for example of f (x) = 3 ?