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Integral ln(x)
(Mathemática | Calculus | Integrals | Table Of | ln x)
 
Discussion of
(integral) ln(x) dx = x ln(x) - x + C.

1. Proof

    Strategy: Use Integration by Parts.

    (integral)ln(x) dx

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

    substitute

    (integral) ln(x) dx = (integral) u dv

    and use integration by parts

    = uv - (integral) v du

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

    = ln(x) x - (integral) x (1/x) dx
    = ln(x) x - (integral) dx
    = ln(x) x - x + C
    = x ln(x) - x + C.
    Q.E.D.

  
 
  

 
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