Can we find the antiderivative of $x^x$? I know that $\int x^xdx$ is not an elementary expression, but can we find another expression？
Asked
Active
Viewed 80 times
0
-
Try with $\displaystyle \int_1^x t^tdt$. – azif00 Oct 29 '19 at 04:29
1 Answers
2
This is the Somophore's dream.
Whet you can do is to write $$x^x=e^{x\log(x)}=\sum_{n=0}^\infty \frac{x^n \log^n(x)}{n!}$$ and face integrals $$I_n=\int x^n \log^n(x)\,dx$$ which write $$I_n=\log ^{n+1}(x) (-E_{-n}(-(n+1) \log (x)))$$ where appear exponential integral functions.
Claude Leibovici
- 260,315