Integral of $$$\frac{1}{2 x}$$$

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

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int \frac{1}{2 x}\, dx = \frac{\ln\left(\left|{x}\right|\right)}{2} + C$$$A