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

Find $$$\int \frac{1}{2 y}\, dy$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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