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

Find $$$\int \left(- \frac{2}{y}\right)\, dy$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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