Calculate :
Split the integrals into:
$\displaystyle \int \frac{2}{x}\ln{(x)}~dx + \int \frac{1}{2}(\ln{(x)})^2~dx$
First integral is dealt with by the substitution $\displaystyle u = \ln{x}$
$\displaystyle \int \frac{1}{2}(\ln{(x)})^2~dx = \int \frac{1}{2}(x)'(\ln{(x)})^2~dx$
Second integral requires the use of integration by parts with $\displaystyle u = (\ln{(x)})^2 $ and $\displaystyle v = x$