Derivation and Evaluation
The steps to calculate the integral of the natural logarithm function:
\[ \int \ln x \, dx \]are presented below.
We first rewrite the given integral as:
\[ \int \ln x \, dx = \int 1 \cdot \ln x \, dx \]Let \( u' = 1 \) and \( v = \ln x \), whose first derivatives are given by \( u = x \) and \( v' = \dfrac{1}{x} \).
Our integral is of the form:
\[ \int \ln x \, dx = \int u' \cdot v \, dx \]Use integration by parts formula \( \int u' v \, dx = u v - \int u v' \, dx \) to write:
\[ \int \ln x \, dx = u v - \int u \cdot v' \, dx \]Substitute \( u, v, \) and \( v' \) to obtain:
\[ = x \ln x - \int x \cdot \dfrac{1}{x} \, dx \]Simplify the term on the right:
\[ = x \ln x - \int 1 \, dx \]Evaluate the integral:
\[ = x \ln x - x + c \]where \( c \) is the constant of integration.
Integral Formula for \( \ln x \):
\[ \int \ln x \, dx = x \ln x - x + c \]