Compute $\int_0^1 \left( \Gamma(x+1) - x^x\right) dx$

Question

Compute

$$\int_0^1 \left( \Gamma(x+1) - x^x\right) dx$$

... Where do I start. I can get the second term in series. $\Gamma(x+1)$ is the "continuous" factorial.

Answer 1

Both $\Gamma(x+1)$ and $x^x$ are log-convex on $(0,1)$, hence almost any numerical method is able to produce accurate approximations of such integral. A tailor-made approach comes from noticing that $\Gamma(x+1)-x^x$ approximately behaves like $\frac{1}{2}\sqrt{x}(1-x)$ on $(0,1)$, hence very accurate approximations are expected from enforcing the substitution $x=u^2$, then applying Newton-Cotes formulas of moderately high order, like Boole's rule.

A closed form is hopeless, if not in terms of a fast-convergent series.