Integrating Ln(x) with Two Easy Methods

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Integrating ln(x) can be effectively accomplished using two methods: partial integration and differentiation with respect to a parameter. The first method employs Leibnitz's rule, leading to the integral of ln(x) being expressed as x ln(x) - x plus a constant. The second method involves integrating the function x^p, differentiating both sides with respect to p, and then setting p to zero to find the integral. Both approaches provide a clear path to solving the integral of ln(x). Understanding these methods enhances integration skills in calculus.
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how do you intergrate lnx?
 
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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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