Question

Find the integral of $\ln x$.

Answer 1

Find the integral of the given function

Given function: $\ln \left(x\right)$

We know that,

$\int udv=uv-\int vdu$

Let ,$u=\ln \left(x\right)$ and $dv=dx$

$\Rightarrow du=\frac{1}{x}dx$ and $v=x$

So,

$$\begin{gathered} \int \ln \left(x\right)dx=\ln \left(x\right)x-\int x.\frac{1}{x}dx \\ =x\ln \left(x\right)-\int dx \end{gathered}$$

$=x\ln \left(x\right)-x+C$, where $C$ is the integration constant.

Hence, the integral of $\ln x$ is $x\ln \left(x\right)-x+C$ where $C$ is the integration constant.