# Integral of Natural Logarithm : ln x

\( \) \( \)\( \)\( \)\( \)\( \)The steps to calculate the integral of the natural logarithm function : \( \displaystyle \int \ln x \; dx \) are presented.

We first rewrite the given intergral as

\( \displaystyle \int \ln x \; dx = \int 1 \cdot \ln x \; dx \)

Let \( u = x \) and \( v = \ln x \) whose first derivatives are given by \( u' = 1 \) and \( v' = \dfrac{1}{x} \)

Our integral is of the form

\( \displaystyle \int \ln x \; dx = \int u' \cdot v \; dx \)

Use the integration by parts to write

\( \displaystyle \int \ln x \; dx = u v - \int u \cdot v' \; dx \)

Substitute \( u, v \) and \( v' \) to obtain

\( \displaystyle = x \ln x - \int x \dfrac {1}{x} \; dx \)

Simplify the term on the right

\( \displaystyle = x \ln x - \int dx \)

and evaluate the integral

\( = x \ln x - x + c \)

where \( c \) is the constant of integration.

Hence

\[ \displaystyle \int \ln x \; dx = x \ln x - x + c \]

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