Integral of $$$- \frac{1}{x}$$$

Find $$$\int \left(- \frac{1}{x}\right)\, dx$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int \left(- \frac{1}{x}\right)\, dx = - \ln\left(\left|{x}\right|\right) + C$$$A