Integral of $$$\ln\left(x\right)$$$

Find $$$\int \ln\left(x\right)\, dx$$$.

For the integral $$$\int{\ln{\left(x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=\ln{\left(x \right)}$$$ and $$$\operatorname{dv}=dx$$$.

Then $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (steps can be seen here) and $$$\operatorname{v}=\int{1 d x}=x$$$ (steps can be seen here).

Thus,

$$\color{red}{\int{\ln{\left(x \right)} d x}}=\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}=\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$x \ln{\left(x \right)} - \color{red}{\int{1 d x}} = x \ln{\left(x \right)} - \color{red}{x}$$

Therefore,

$$\int{\ln{\left(x \right)} d x} = x \ln{\left(\left|{x}\right| \right)} - x$$

Simplify:

$$\int{\ln{\left(x \right)} d x} = x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)$$

Add the constant of integration:

$$\int{\ln{\left(x \right)} d x} = x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)+C$$

Answer: $$$\int{\ln{\left(x \right)} d x}=x \left(\ln{\left(\left|{x}\right| \right)} - 1\right)+C$$$