# Solve this integral

1. Nov 6, 2005

### latyph

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how do i get to solve this integral,i have no idea whatsoever so no one should expect what i have done.it was presented to me by a colleague
[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]............

2. Nov 6, 2005

### HallsofIvy

Staff Emeritus
Do you mean the integral of that function?
Do you have any reason to believe that function is well-defined, much less integrable?

A couple of points: since $[x^a]^b= x^{ab}$, $[x^{\frac{1}{x}}]^{x^{\frac{1}{x}}}= x^{\frac{2}{x}}$. In general then, that "stack" of $x^{\frac{1}{x}}$, n times, is the same as $x^{\frac{n}{x}}$ and I see no reason to think that expanding it to infinity will give a function.

3. Nov 6, 2005

### benorin

Power Tower

This is not well defined, it seems. Consider that $e^{x^{2}}$ is not the same as $\left(e^{x}\right)^{2}=e^{2x}$. The one way to look at the given function is as a power tower (e.g. $$x^{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}}$$,) see http://mathworld.wolfram.com/PowerTower.html for a reference; and another way is as HallsofIvy pointed out. I suppose it would depend on how, that is, by what limiting process, the given integrand is being defined. You might try defining the function as a limit of a sequence of functions, perhaps you can use the Lebesgue's Dominated Convergence Theorem to show convergence of the integral (supposing it's a definite one).

4. Nov 6, 2005

### fourier jr

i get the feeling that's a bull**** integral that the 'colleague' gave out....

5. Nov 6, 2005

### benorin

Why? Do you suppose it is homework?

6. Nov 7, 2005

### latyph

But Cant The Function Be Resolved To A Definite One That Can Be Integrable