Integrating The Infinite Power Tower

Calculus Level 5

\[\Large f(x)=x^{x^{x^{x^{\dots}}}}\]

The function above is what is known as the infinite power tower. At first glance, the function above may seem like it quickly spirals off to infinity, but for select values of \(x\), surprisingly the function converges. If \(x\in[e^{-e},e^{\frac{1}{e}}]\), then \(f(x)\) is defined and converges to a finite number. What is the value of the integral below? Give your answer to 3 decimal places.

\[\Large \displaystyle\int_{e^{-e}}^{e^{\frac{1}{e}}} f(x)\, dx\]

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