How do you intergrate ln(x)?

There's a nice trick here you can do, treat the equation as 1*ln(x) then intergrate by parts.

Differentiating ln(x) gives 1/x, while intergrating 1 gives x

So your left with a much easier intergration

xln(x)-(Intergral sign)x 1/x dx

which is simply x*ln(x)-x