Evaluate the integral.

${\displaystyle \underset{1}{\overset{e^3}{}}}\frac{ln(x)}{x}dx$

Solution

We let $u=ln(x)$ because its differential $du=\frac{dx}{x}$ occurs in the integral. When $x=1,u=ln(1)=,$ and when $x=e^3,u=ln(e^3)=3.$ Thus,

${\displaystyle \underset{1}{\overset{e^3}{}}}\frac{ln(x)}{x}dx={\displaystyle \underset{0}{\overset{3}{}}}$udu

$=$

$=$