I've never worked with this kind of problem before. Can anyone please provide a step-by-step answer?
let $\displaystyle u = (\ln 5x)^{2} $ and $\displaystyle dv = dx$
then $\displaystyle du = 2 \ln (5x)*\frac{5}{x} \ dx $ and $\displaystyle v =x $
$\displaystyle \int (ln 5x)^{2} = x (\ln 5x)^{2} - 10 \int \ln 5x \ dx $
Integrating by parts again, you can show that $\displaystyle \int \ln 5x = \frac{1}{5}x \ln 5x -\frac{1}{5}x +C $
so $\displaystyle \int (ln 5x)^{2} = x (\ln 5x)^{2} - 2x \ln 5x + 2x + C $