Integral of $$$\frac{3}{2 n}$$$

Find $$$\int \frac{3}{2 n}\, dn$$$.

Apply the constant multiple rule $$$\int c f{\left(n \right)}\, dn = c \int f{\left(n \right)}\, dn$$$ with $$$c=\frac{3}{2}$$$ and $$$f{\left(n \right)} = \frac{1}{n}$$$:

$${\color{red}{\int{\frac{3}{2 n} d n}}} = {\color{red}{\left(\frac{3 \int{\frac{1}{n} d n}}{2}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\frac{1}{n} d n}}}}{2} = \frac{3 {\color{red}{\ln{\left(\left|{n}\right| \right)}}}}{2}$$

Therefore,

$$\int{\frac{3}{2 n} d n} = \frac{3 \ln{\left(\left|{n}\right| \right)}}{2}$$

Add the constant of integration:

$$\int{\frac{3}{2 n} d n} = \frac{3 \ln{\left(\left|{n}\right| \right)}}{2}+C$$

$$$\int \frac{3}{2 n}\, dn = \frac{3 \ln\left(\left|{n}\right|\right)}{2} + C$$$A