Discussion of ln(x) dx = x ln(x) - x + C.

1. Proof

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.
Q.E.D.

 
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