How to integrate $x^{x^x}$

Question

Find $\int_{0}^{1} x^{x^x} dx$.

I cannot show any working as such since I don't really know how to even begin with. This is from a Facebook group : Art of Mathematics.

Answer 1

If you are patient, you could develop $x^{x^x}$ as a series built around $x=0$ and get $$x^{x^x}=x+x^2 \log ^2(x)+\frac{1}{2} x^3 \left(\log ^4(x)+\log ^3(x)\right)+\frac{1}{6} x^4 \left(\log ^6(x)+3 \log ^5(x)+\log ^4(x)\right)+\frac{1}{24} x^5 \left(\log ^8(x)+6 \log ^7(x)+7 \log ^6(x)+\log ^5(x)\right)+\frac{1}{120} x^6 \left(\log ^{10}(x)+10 \log ^9(x)+25 \log ^8(x)+15 \log ^7(x)+\log ^6(x)\right)+O\left(x^7\right)$$ and notice that $$\int_0^1 x^n \log^m(x)\,dx=e^{i \pi m} \frac{ \Gamma (m+1)}{(n+1)^{m+1} }\qquad \text{if} \qquad \Re(n)>-1\land \Re(m)>-1$$ Applying this to the expansion you should get $$\int_{0}^{1} x^{x^x}\, dx=\frac{1062182257609343089}{1853320108689000000}\approx 0.573124$$

Edit

Just for the fun of it, using the same level of expansion $O\left(x^7\right)$, I obtained $$\int_0^1 x^x\, dx=\frac{24659496552164597077}{31476303632793600000}\approx 0.783431$$ $$\int_0^1x^{x^{x^x}}\, dx=\frac{50936874284941789413451325411}{69649555988586178805760000000}\approx 0.731331$$ $$\int_0^1x^{x^{x^{x^x}}}\, dx=\frac{4246847580780746527299311}{7107097549855732531200000}\approx 0.597550$$ while numerical integration would give $0.783431$, $0.731340$ and $0.597578$.