Integral ln x dx

Strategy: Use Integration by Parts.
ln(x) dx

set
u = ln(x), dv = dx
then we find
du = (1/x) dx, v = x

substitute

ln(x) dx = u dv

and use integration by parts

= uv - v du

substitute u=ln(x), v=x, and du=(1/x)dx

= ln(x) x - x (1/x) dx
= ln(x) x - dx
= ln(x) x - x + C
= x ln(x) - x + C.