Answer
$\frac{1}{2}((\ln(2e))^{2})-(\ln(2)^2)$ Or $\approx1.193$
Work Step by Step
$\int_{1}^{e}\frac{\ln(2x)}{x}dx$
Use u-subsitution.
$\frac{du}{dx}= \ln(2x)$
$\frac{du}{dx}=\frac{1}{2x}(2)$
$dx=xdu$
$\int_{1}^{e}\frac{ux}{x}du$
$\int_{1}^{e}udu$
$\frac{u^{2}}{2}|_{1}^{e}$
$\frac{(\ln(2x))^{2}}{2}|_{1}^{e}$
$(\frac{(\ln(2e))^{2}}{2})-(\frac{(\ln(2))^{2}}{2})$
$\frac{1}{2}((\ln(2e))^{2})-(\ln(2)^2)$ Or $\approx1.193$
