# What is the antiderivative of lnx?

Mar 16, 2018

$\int \ln x \mathrm{dx} = x \left(\ln x - 1\right) + \text{c}$

#### Explanation:

To find an antiderivative of $\ln x$, we must find $\int \ln x \mathrm{dx}$. To do so, we use integration by parts.

$\int u \mathrm{dv} = u v - \int v \mathrm{du}$

Let $u = \ln x \Rightarrow \mathrm{du} = \frac{1}{x} \mathrm{dx}$

And $\mathrm{dv} = \mathrm{dx} \Rightarrow v = x$

So

$\int \ln x \mathrm{dx} = x \ln x - \int \mathrm{dx} = x \ln x - x = x \left(\ln x - 1\right)$