You were right so far, now all you have to do is transpose (solve) the equation for int[(lnx)/x]
Notice that you had
int[(lnx)/x]dx = (lnx)^2 - int[(lnx)/x]dx + C
solving for int[(lnx)/x]dx, we obtain:
2*int[(lnx)/x]dx = (lnx)^2 + C
divide by 2 on both sides:
int[(lnx)/x] dx = [(lnx)^2]/2 + C
You should make a note of this technique, it comes in handy when you're integrating things that end up going in a loop and coming back to the problem you started in the first place. Functions like e^x * (any trig function) (for instance, the same thing happens if you try to do int(e^x*sin(x) dx) by parts) do something like this when you try to integrate them by parts, so all you do is just solve for the thing you want, don't keep integrating by parts over and over.