If the shortest distance between the lines (x+2)/2=(y+3)/3=(z-5)/4 and (x-3)/1=(y-2)/-3=(z+4)/2 is 38/(3 √ 5) k, and ∫_0^k [x^2] d x=α-√ α, where [x]…

If the shortest distance between the lines $\displaystyle \frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\displaystyle \frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\displaystyle \frac{38}{3 \sqrt{5}} \mathrm{k}$, and $\displaystyle \int_0^{\mathrm{k}}\left[x^2\right] \mathrm{d} x=\alpha-\sqrt{\alpha}$, where $\displaystyle [x]$ denotes the greatest integer function, then $\displaystyle 6 \alpha^3$ is equal to $\displaystyle \_\_\_\_$.