Integral of $$$\frac{a}{v}$$$ with respect to $$$v$$$

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

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

$${\color{red}{\int{\frac{a}{v} d v}}} = {\color{red}{a \int{\frac{1}{v} d v}}}$$

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

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

Therefore,

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

Add the constant of integration:

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

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