Mathematics - Vector Question with Solution | TestHub

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}\mathrm{k}$, and ${\int }_0^{\mathrm{k}}\left[x^2\right]\mathrm{d}x=\alpha -\sqrt{\alpha }$, where $[x]$ denotes the greatest integer function, then $6{\alpha }^3$ is equal to________