Why Can't We Integrate e^(x^2) Using Elementary Functions?

[kmr159]

Homework Statement

integrate e^(x2)

Homework Equations

The Attempt at a Solution

e^(x2) = (e^x)²

substitute (e^x) = u so (e^x)dx = du

therefore

∫e^(x2) dx = ∫u du

Unfortunately this is not the correct answer
Can someone please tell me what I am doing wrong?

Thanks

 

[STEMucator]

Do you know the Taylor expansion of $e^x$?

 

[Zeda]

The first line of your attempt is where you went wrong. $e^{(x^{2})}\neq (e^{x})^{2}$. Rather, $(e^{x})^{2}=e^{x}e^{x}=e^{2x}$

Otherwise, I kind of suck at this kind of problem. I never remembered all the fun rules, so I would go the really long route:
$e^{(x^{2})}=1+x^{2}+x^{4}/2+x^{6}/6+x^{8}/24+x^{10}/120+...$
so the integral would be
$x+x^{3}/3+x^{5}/10+x^{7}/42+x^{9}/216+x^{11}/1320+...$

From there, I guess good luck?

 

[Dick]

Homework Statement

integrate e^(x2)

Homework Equations

The Attempt at a Solution

e^(x2) = (e^x)²

substitute (e^x) = u so (e^x)dx = du

therefore

∫e^(x2) dx = ∫u du

Unfortunately this is not the correct answer
Can someone please tell me what I am doing wrong?

Thanks

You can only express the integral of $e^{(x^2)}$ in term of nonelementary functions like the error function 'erf'. http://en.wikipedia.org/wiki/Error_function Is that what you are expected to do?

 

[Ray Vickson]

Homework Statement

integrate e^(x2)

Homework Equations

The Attempt at a Solution

e^(x2) = (e^x)²

substitute (e^x) = u so (e^x)dx = du

therefore

∫e^(x2) dx = ∫u du

Unfortunately this is not the correct answer
Can someone please tell me what I am doing wrong?

Thanks

To expand on Dick's answer: it has been rigorously shown that it is impossible to express the indefinite integral of $\exp(x^2)$ in terms of a finite number of elementary functions. It is not just that nobody has been smart enough to figure out how to do it; it is proven that is it impossible for anybody to do, ever. Even if you write trillions of terms on a piece of paper as large as the solar system you still cannot write out the result exactly. Of course, you can express the result in non-finite terms, such as through an infinite series, etc.

 

[Dick]

To expand on Dick's answer: it has been rigorously shown that it is impossible to express the indefinite integral of $\exp(x^2)$ in terms of a finite number of elementary functions. It is not just that nobody has been smart enough to figure out how to do it; it is proven that is it impossible for anybody to do, ever. Even if you write trillions of terms on a piece of paper as large as the solar system you still cannot write out the result exactly. Of course, you can express the result in non-finite terms, such as through an infinite series, etc.

Nice expansion, Ray Vickson. I was just interested in what was expected. But that gave it more depth.