∫lnx/(x-2)^2dx
=-∫lnxd[1/(x-2)]
=-lnx/(x-2)+∫1/(x-2)*dlnx
=-lnx/(x-2)+∫1/[x(x-2)]dx
=-lnx/(x-2)+1/2∫[1/(x-2)-1/x]dx
=-lnx/(x-2)+1/2∫1/(x-2)dx-1/2∫1/xdx
=-lnx/(x-2)+1/2ln(x-2)-1/2lnx+C
=-lnx/(x-2)+1/2ln[(x-2)/x]+C