Proving Integral of x^x Does Not Exist

In summary, the conversation discusses the function x^x and how it has no elementary function to define its antiderivative. Various resources, including a 1994 article and a 1948 book, are mentioned as references for further understanding. The possibility of using e^{x\ln x} as a way to fit the theorem for x^x is also mentioned.
  • #1
n1person
145
0
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?
 
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  • #2


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


Does anyone happen to have a more accessible resource about this subject? Like something online?
 
  • #5


so what we have is:

[tex]
\int f(x)e^g^(^x^)dx
[/tex]
[tex]
g(x)=0
[/tex]
[tex]
f(x)=x^x
[/tex]

so the formula given

[tex]

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

[/tex]

just goes to

[tex]
f(x)=R'(x)
[/tex]

which isn't overly illuminating :(
 
  • #6


n1person said:
so what we have is:

[tex]
\int f(x)e^g^(^x^)dx
[/tex]
[tex]
g(x)=0
[/tex]
[tex]
f(x)=x^x
[/tex]

so the formula given

[tex]

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

[/tex]

just goes to

[tex]
f(x)=R'(x)
[/tex]

which isn't overly illuminating :(

No, [itex]x^x[/itex] is not a rational function, so this is not the way to fit the theorem.
In fact, the case of [itex]x^x[/itex] (as explained in the actual references) is a bit more involved than that simplistic web page says.
 
  • #7


How about letting [tex] x^x = e^{x\ln x} [/tex]?
 

1. What is the integral of x^x?

The integral of x^x is a mathematical expression that represents the area under the curve of the function x^x. It is denoted by ∫x^x and is an important concept in calculus.

2. How do you prove that the integral of x^x does not exist?

To prove that the integral of x^x does not exist, we can use the Cauchy's convergence test or the limit comparison test. These tests involve taking the limit of the function as x approaches infinity to determine if the integral converges or diverges.

3. Can the integral of x^x be evaluated using traditional integration techniques?

No, the integral of x^x cannot be evaluated using traditional integration techniques such as integration by parts or substitution. This is because x^x does not have an elementary antiderivative.

4. Are there any alternate methods for evaluating the integral of x^x?

Yes, there are alternative methods for evaluating the integral of x^x, such as using numerical integration techniques like Simpson's rule or the trapezoidal rule. These methods involve approximating the area under the curve using smaller, simpler shapes.

5. Why is it important to prove that the integral of x^x does not exist?

Proving that the integral of x^x does not exist helps us understand the behavior of this function and its limitations. It also allows us to use other methods for evaluating the integral and find ways to approximate its value, which can be useful in various applications of calculus.

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