Integral of $$$\frac{2}{v}$$$

Find $$$\int \frac{2}{v}\, dv$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int \frac{2}{v}\, dv = 2 \ln\left(\left|{v}\right|\right) + C$$$A