Use integration by parts to find the integral:
Integration by parts;
$\displaystyle \int u \, dv=uv - \int v \, du$
Function:
$\displaystyle \int(ln(4x))^2$
let $\displaystyle u=(ln(4x))^2$
$\displaystyle dv=dx$
$\displaystyle du=2(ln(4x))\frac{1}{4x}$
$\displaystyle v=x$
=$\displaystyle x(ln(4x))^2-\int2x(ln(4x))\frac{1}{4x}$
Then simplify the remaining integral and integrate once again