The Integral of ln (x) and (ln (x))^n

The natural logarithm ln (x) and its inverse function e^(x) have special derivatives which must be memorized because they do not fall under a more general differentiation rule. e^x is easy because its derivative and antiderivative are itself. The derivative of ln (x) is 1/x, but its integral is often left as an exercise in a section about integration by parts. It is generally the first instance of integration by parts where one of the two functions (u and dv or f and g') is the invisible unity, one.

The Integral of ln (x):

IBP Formula

We let u = ln (x) , so du =1/x
And v = x because we let dv = 1dx

The IBP formula becomes:

Thus:

When we take this same approach to the antiderivative of ln^k (x) we begin to see a pattern which yields a simplified integral equation:

First application of integration by parts:

Second application of integration by parts:



And as the pattern continues, we see that: