Integral of $$$\frac{x - y}{y}$$$ with respect to $$$y$$$

Find $$$\int \frac{x - y}{y}\, dy$$$.

Expand the expression:

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dy = c y$$$ with $$$c=1$$$:

$$\int{\frac{x}{y} d y} - {\color{red}{\int{1 d y}}} = \int{\frac{x}{y} d y} - {\color{red}{y}}$$

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

$$- y + {\color{red}{\int{\frac{x}{y} d y}}} = - y + {\color{red}{x \int{\frac{1}{y} d y}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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