# The “natural” Sophomore's Dream integral: $\int_{0}^{\infty} x^{-x}\ dx$

I have been wondering about this for a while, but with no real luck in figuring it out. The famous "Sophomore's dream" identity refers to two similar integrals, one of which is

$$\int_{0}^{1} x^{-x}\ dx = \sum_{n=1}^{\infty} n^{-n}$$

which has a pleasing symmetry between integrand and summand. However, I also notice that $$x \mapsto x^{-x}$$ is a rapidly-converging-to-zero function of $$x$$, and hence it seems like it might be more "natural" to then wonder about the integral

$$\int_{0}^{\infty} x^{-x}\ dx$$.

Numerical integration suggests it is approximately 1.99545596. Yet what I would like to find is some other representation that is not directly an integral, whether it be a series, or even a finite expression using any already-established mathematical functions and/or constants.

And it doesn't seem very clear at all how to do this. Obviously, with upper limit $$\infty$$, Taylor expansion of the integrand is of no use since it will only ever have finite radius of convergence. The other line of attack that I thus think of is to try and express $$x^{-x}$$ as a series of some form of decaying functions that are simpler to integrate. We do have that

$$x^{-x} = e^{-x \ln x}$$

but this is of no use: it doesn't yield any series in terms of $$e^{-x}$$-like terms. We have the interesting substitution $$x = e^{W(u)}$$, $$dx = \frac{1}{1 + W(u)}\ du$$ (equiv. to $$u = x \ln x$$) with the Lambert W-function, which gives

$$\int_{0}^{\infty} x^{-x}\ dx = \int_{0}^{\infty} e^{-u} \frac{du}{1 + W(u)}$$

but this doesn't help for series expansions.

• This looks hard.. But atleast we only have to find $\int_1^\infty x^{-x} dx$ since the first part is known. However expanding into power series here is painful. – Nyssa Jun 20 at 9:07
• See my older question How to show that $\int_0^\infty\frac1{x^x}\,dx<2$ regarding the second paragraph. – TheSimpliFire Aug 20 at 12:06
Although it is not exactly for what the OP is asking for, but still may be interesting. Ramanujan provides curious expansion $$\int_{0}^\infty x^{-x}dx \sim \sum_{n \in \mathbb{Z}} n^{-n}$$ (be careful, because the sum is divergent). See Theorem 2 in Berndt, Evans http://math.ucsd.edu/~revans/Elegant.pdf.