Integral of $$$\frac{2}{x}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\frac{2}{x} d x}}} = {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

The integral of $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$2 {\color{red}{\int{\frac{1}{x} d x}}} = 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Therefore,

$$\int{\frac{2}{x} d x} = 2 \ln{\left(\left|{x}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{2}{x} d x} = 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{2}{x}\, dx = 2 \ln\left(\left|{x}\right|\right) + C$$$A