Integral of $$$\frac{e}{t}$$$

Find $$$\int \frac{e}{t}\, dt$$$.

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

$${\color{red}{\int{\frac{e}{t} d t}}} = {\color{red}{e \int{\frac{1}{t} d t}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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