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

Find $$$\int \frac{x - 2}{x}\, dx$$$.

Expand the expression:

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

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{\frac{2}{x} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{2}{x} d x} + {\color{red}{x}}$$

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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