Using my calculator, I had tried to evaluate the definite integral $$\int_{0}^{1}x^x \mathbf{d}x$$ However, according to my CASIO $fx-570$ES, the result was Math ERROR, which led me to believe that the definite integral does not exist at all, or was infinite.
In contrast, Wolframalpha showed different results. Why do the results on both calculators differ? I cannot figure out a reason for the difference.
The function $f(x) = x^x$ for $x > 0$ and $f(0) = 1$ is well-defined, continuous, and bounded between $0.5$ and $1$ on the interval $[0, 1]$, so the integral does exist.
Whether it can be expressed in terms of "nice" constants is a rather different question; in particular, your calculator is likely unhappy with the (apparent but not actually existent) singularity at zero.
As far as computing the integral, you could simply integrate on $[0.1, 1]$ (guaranteed to be correct with an error at most $0.1$), $[0.001, 1]$ (guaranteed to have error at msot $0.001$), and so on; I'm sure that this will get rather close to the Wolfram Alpha result if you choose a small enough threshold.
As a fun fact, this is related is the so-called sophomore's dream which gives a series representation for the integral, roughly around $0.783$.