What is the integral of ln(x)?

integral of ln(x)

The integral of

\ln(x)

x\ln(x) - x + C

.

Solution. We want to determine the integral of

\ln(x)

\begin{align*} \int \ln(x) dx. \end{align*}

We will integrate by parts, i.e.,

\begin{align*} \int UdV = UV - \int VdU, \end{align*}

where

\begin{align*} U = \ln(x), \quad &dV = dx\\ dU = \frac{1}{x}dx, \quad &V = x. \end{align*}

The reader can verify that it is correct for

dU

here, and for

V

it is straightforward. So we get the following:

\begin{align*} \int \ln(x) dx &= x\ln(x) - \int x \cdot \frac{1}{x} dx \\ &= x \ln(x) - \int dx \\ &= x \ln(x) - x + C. \end{align*}

Therefore, the integral of

\ln(x)

x\ln(x) - x + C