# How does one go about proving an elementary solution to an integral does not exist?

1. Aug 12, 2009

### n1person

Recently I have begun thinking about the function x^x. I am well aware that there is no elementary function to define it's antiderivative, and intuitively it makes sense (I cannot think of an elementary function who's derivative is x^x). However, how would one go about proving this rigorously?

2. Aug 12, 2009

### g_edgar

Re: How does one go about proving an elementary solution to an integral does not exis

MARCHISOTO and ZAKERI (1994): "An Invitation to Integration in Finite Terms" , The College Mathematics Journal 25 No. 4 Sept. pp 295 - 308

J. F. Ritt, Integration in finite terms: Liouville's theory of elementary methods, 1948

3. Aug 12, 2009

### n1person

Re: How does one go about proving an elementary solution to an integral does not exis

4. Aug 13, 2009

### Big-T

5. Aug 13, 2009

### n1person

Re: How does one go about proving an elementary solution to an integral does not exis

so what we have is:

$$\int f(x)e^g^(^x^)dx$$
$$g(x)=0$$
$$f(x)=x^x$$

so the formula given

$$f(x)=R'(x)+g(x)R(x)$$

just goes to

$$f(x)=R'(x)$$

which isn't overly illuminating :(

6. Aug 13, 2009

### g_edgar

Re: How does one go about proving an elementary solution to an integral does not exis

No, $x^x$ is not a rational function, so this is not the way to fit the theorem.
In fact, the case of $x^x$ (as explained in the actual references) is a bit more involved than that simplistic web page says.

7. Aug 13, 2009

### Big-T

Re: How does one go about proving an elementary solution to an integral does not exis

How about letting $$x^x = e^{x\ln x}$$?