Have I understood this correctly? Integrating Ln[x]

Hello all!
I am an upper 6th student doing maths and physics (not further maths) going on to do physics at Uni, and was trying to figure out how to integrate Ln[x], here's my method, could someone look over it and see if my reasoning and explanation of my working is clear, logical and actually possible?
Thanks very much

$u = \ln (x) = ln (v)$ ∴ $v = x$

Currently we are integrating everything with respect to x, but we should consider this equation as integrating with respect to v and u, as shown:
$\int udv = uv - \int vdu$

Although your answer is correct, this line makes no sense.

Also, you shouldn't be saying "differential", the correct word is "derivative".



Not really, no. Your first formula of integration by parts was correct. "dv" and "du" don't really make sense by themselves - they're short forms for dv/dx or du/dx that people sometimes use, but you're not really integrating w.r.t u or v. So where you write stuff like

"du = dx/x" - that doesn't strictly make sense either. It's just abusing notation because it still works.

right, I fixed those few things u pointed out, thanks
du = dx/x, you say it doesnt strictly work... however I can use that as a means for finding the integral of ln[x] correct?
Thanks very much