Question

${\int }_{\frac{1}{e}}^e\frac{dx}{x{\left(\log x\right)}^{\frac{1}{3}}}$

Answer 1

${\int }_{1/e}^l\frac{dx}{{x\left(\log x\right)}^{\frac{1}{3}}}$

$\log x=t$ when $x=e$ $t=1$

$\frac{dx}{x}=dtx=\frac{1}{e}t=-1$

$={\int }_{-1}^1\frac{dt}{{\left(t\right)}^{\frac{1}{3}}}={\int }_{-1}^1\left(t^{\frac{-1}{3}}\right)dt=\frac{3}{2}{\left[t^{\frac{2}{3}}\right]}_{-1}^1=\frac{3}{2}[1-1]$

$=0$