Integral of $$$\frac{r}{a e^{2}}$$$ with respect to $$$a$$$

Find $$$\int \frac{r}{a e^{2}}\, da$$$.

Apply the constant multiple rule $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ with $$$c=\frac{r}{e^{2}}$$$ and $$$f{\left(a \right)} = \frac{1}{a}$$$:

$${\color{red}{\int{\frac{r}{a e^{2}} d a}}} = {\color{red}{\frac{r \int{\frac{1}{a} d a}}{e^{2}}}}$$

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

$$\frac{r {\color{red}{\int{\frac{1}{a} d a}}}}{e^{2}} = \frac{r {\color{red}{\ln{\left(\left|{a}\right| \right)}}}}{e^{2}}$$

Therefore,

$$\int{\frac{r}{a e^{2}} d a} = \frac{r \ln{\left(\left|{a}\right| \right)}}{e^{2}}$$

Add the constant of integration:

$$\int{\frac{r}{a e^{2}} d a} = \frac{r \ln{\left(\left|{a}\right| \right)}}{e^{2}}+C$$

$$$\int \frac{r}{a e^{2}}\, da = \frac{r \ln\left(\left|{a}\right|\right)}{e^{2}} + C$$$A