When I was messing about with integrals, I came to know about this result: $$f(x) = \int_0 ^x \left(\frac{x}{t}\right)^t \mathrm dt = \sum _{n=1}^\infty \left(\frac{x}{n}\right)^n$$ and the related but just slightly less beautiful $$g(x) = \int_0 ^x \left(\frac{t}{x}\right)^t \mathrm dt = \sum _{n=1}^\infty (-1)^{n+1} \left(\frac{x}{n}\right)^n.$$ These results can be found by keeping the things inside the integrals in the form $e^{\ln k(x)}$, writing out the Taylor series of $e^x$, and then finally integrating by parts (that's how I found them) or you can do some u-substitutions to find the gamma function in here. Are these functions special? Can the integrals be evaluated? If not, can other non-elementary integrals be expressed in terms of this function? I tried to evaluate the integrals in terms of other functions, but I was unsuccessful. I do not think this has an elementary solution.
Edit: I realized that $g(x)$ can be written kind of like $f(x)$. $$g(x) = \int_0 ^x \left(\frac{t}{x}\right)^t \mathrm dt = \int_0 ^x \left(\frac{x}{t}\right)^{-t} \mathrm dt$$ Substituting $u=-t$, we have $$g(x) = -\int_0 ^{-x} \left(\frac{x}{t}\right)^t \mathrm dt$$ So we get $$\int_0 ^x \left(\frac{x}{t}\right)^t \mathrm dt + \int_0 ^{-x} \left(\frac{x}{t}\right)^t \mathrm dt = \sum_{n=1} ^{\infty}\left(\frac{x}{2n+1}\right)^{2n+1}$$ and $$\int_0 ^x \left(\frac{x}{t}\right)^t \mathrm dt - \int_0 ^{-x} \left(\frac{x}{t}\right)^t \mathrm dt = \sum_{n=1} ^{\infty}\left(\frac{x}{2n}\right)^{2n}$$ Which is pretty cool.
Edit: Here's something to think about. What happens if we use the Euler-Maclaurin Summation formula on the case where $x=1$? What if we use it on any $x$?
