Write a value

$y=\int 1 \times \log _{e} x d x$

By using integration by parts

Let, $\log _{e} x$ as Ist function and 1 as IInd function

Use formula $\int I \times I I d x=I \int I I d x-\int\left(\frac{d}{d x} I\right)\left(\int I I d x\right) d x$

$y=\log _{e} x \int d x-\int\left(\frac{d}{d x} \log _{e} x\right)\left(\int d x\right) d x$

$y=\left(\log _{e} x\right) x-\int\left(\frac{1}{x}\right)(x) d x$

$y=x \log _{e} x-\int d x$

$y=x \log _{e} x-x+c$