Question

Write a value of $\int {\log }_{e}xdx$.

Answer 1

$\int$log_e x dx
= $\int$ $\underset{\mathrm{II}}{1}.\underset{\mathrm{I}}{{\log }_{e}x}$ dx
=${\log }_{e}x\int 1dx-\int \left\{\frac{d}{dx}\left({\log }_{e}x\right)\int 1dx\right\}dx$
= log_e x $\int$1 . dx – ​$\int$ $\frac{1}{x}\times x$.dx
= log_e x × x – ​$\int$ dx
=​ x log_e x – x + C
=​ x log_e x – x + C
= x (log_e x – 1) + C