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.
Now Put Limits
[ln(1)-1+C]-[0-0+C]= -1
Note-Lim [xlnx] = 0.
x->0