Can the integral $\int x^x \, \mathrm{d}x$ be evaluated?

Question

It's obvious that $\dfrac{d(x^x)}{dx}=x^x (1+\ln x)$, then what about its integral? Special functions can be used if this integral can't be evaluated with simple functions, e.g. Gamma function $\Gamma(x)$ or Lambert W-function $W(x)$.

Thanks.

Answer 1

To answer the question, this function can not be integrated in terms of elementary functions as shown by @shrey's comment. So there is no "simple" answer to your question, unless you are willing to consider a series approximation:

$$\int{x^xdx} = \int{e^{\ln x^x}dx} = \int{\sum_{k=1}^{\infty}\frac{x^k\ln^k x}{k!}}dx$$ Hope it helps.