Integrating Ln(x) with Two Easy Methods

[matt_crouch]

how do you intergrate lnx?

 

[Count Iblis]

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!