Integral of $$$\frac{6}{r}$$$

Find $$$\int \frac{6}{r}\, dr$$$.

Apply the constant multiple rule $$$\int c f{\left(r \right)}\, dr = c \int f{\left(r \right)}\, dr$$$ with $$$c=6$$$ and $$$f{\left(r \right)} = \frac{1}{r}$$$:

$${\color{red}{\int{\frac{6}{r} d r}}} = {\color{red}{\left(6 \int{\frac{1}{r} d r}\right)}}$$

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

$$6 {\color{red}{\int{\frac{1}{r} d r}}} = 6 {\color{red}{\ln{\left(\left|{r}\right| \right)}}}$$

Therefore,

$$\int{\frac{6}{r} d r} = 6 \ln{\left(\left|{r}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{6}{r} d r} = 6 \ln{\left(\left|{r}\right| \right)}+C$$

$$$\int \frac{6}{r}\, dr = 6 \ln\left(\left|{r}\right|\right) + C$$$A