Question

Integration of $\ln xdx$ is

Answer 1

Step 1: Assume the variables

To find $\int \ln \left(x\right)dx$, we can use the integration by parts method.

$\int udv=uv-\int vdu$

Let us take, $u=\ln \left(x\right)$ and

$$\begin{gathered} dv=dx \\ \Rightarrow v=x \end{gathered}$$

Step 2: Substitute the variables

Now, apply the values in the formula and integrate the function,

Hence,

$$\begin{gathered} \int \ln \left(x\right)dx=x\ln \left(x\right)-\int x\frac{d}{dx}\left(\ln \right(x\left)\right)dx \\ =x\ln \left(x\right)-\int x\times \left[\frac{1}{x}\right]dx \\ =x\ln \left(x\right)-\int dx \\ =x\ln \left(x\right)-x+C \end{gathered}$$

where $C$ is the constant of integration.

Hence, the integration of $\int \ln \left(x\right)dx$ is $x\ln \left(x\right)-x+C$.