Integral of $$$\frac{4}{y}$$$

Find $$$\int \frac{4}{y}\, dy$$$.

Apply the constant multiple rule $$$\int c f{\left(y \right)}\, dy = c \int f{\left(y \right)}\, dy$$$ with $$$c=4$$$ and $$$f{\left(y \right)} = \frac{1}{y}$$$:

$${\color{red}{\int{\frac{4}{y} d y}}} = {\color{red}{\left(4 \int{\frac{1}{y} d y}\right)}}$$

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

$$4 {\color{red}{\int{\frac{1}{y} d y}}} = 4 {\color{red}{\ln{\left(\left|{y}\right| \right)}}}$$

Therefore,

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{4}{y} d y} = 4 \ln{\left(\left|{y}\right| \right)}+C$$

$$$\int \frac{4}{y}\, dy = 4 \ln\left(\left|{y}\right|\right) + C$$$A