Question

Evaluate: ${\int }_e^{e^2}\frac{dx}{x\log x}.$

Answer 1

Let $I={\int }_e^{e^2}\frac{1}{xlogx}dx$

$\Rightarrow {\int }_e^{e^2}\left(\frac{1}{x}\right)\left(\frac{1}{logx}\right)dx$

Let $logx=u\Rightarrow du=\frac{1}{x}dx$

Then $I={\int }_e^{e^2}\frac{1}{u}du$

$\Rightarrow ln[ln|x|{]}_e^{e^2}+C$

$=ln\left|ln{e}^2\right|-ln\left|lne\right|=ln2$