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

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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