Question

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

Answer 1

${\int }_{\frac{1}{e}}^e\frac{dx}{x910gx}\frac{1}{3}$

put log x = t when x=e t=1

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

${\int }_{-1}^1\frac{dt}{t\frac{1}{3}}$

$=[\frac{t^{\frac{-1}{3}+1}}{\frac{-1}{3}+1}{]}_{-1}^1$

$=\frac{(t^{\frac{2}{3}}{)}_{-1}^1}{\frac{2}{3}}$

$=\frac{1}{\frac{2}{3}}-\frac{1}{\frac{2}{3}}=0$