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How do you intergrate lnx?

  1. Mar 30, 2009 #1
    how do you intergrate lnx?
  2. jcsd
  3. Mar 30, 2009 #2
    Re: intergratation

    There are two easy ways:

    1) Partial integration

    2) Differentiation w.r.t. a parameter.

    Method 1):

    Leibnitz's rule can be written as:

    d(fg) = f dg + g df

    We can rewrite this as:

    f dg = d(fg) - g df (1)

    So, if we want to integrate ln(x), we can write according to Eq. (1):

    ln(x) dx = d[x ln(x)] - x d[ln(x)] = d[xln(x)] - dx

    So, the integral of ln(x) dx is the integral of d[xln(x)] minus the integral of dx, which is x ln(x) - x plus an arbitrary constant.

    Method 2):

    Consider integrating the function x^p. Then differentiate both sides w.r.t. the parameter p. Then set p equal to zero. Try it!
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