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

Find $$$\int \left(- \frac{1}{t}\right)\, dt$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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