Integral of $$$e r^{3}$$$

Find $$$\int e r^{3}\, dr$$$.

Apply the constant multiple rule $$$\int c f{\left(r \right)}\, dr = c \int f{\left(r \right)}\, dr$$$ with $$$c=e$$$ and $$$f{\left(r \right)} = r^{3}$$$:

$${\color{red}{\int{e r^{3} d r}}} = {\color{red}{e \int{r^{3} d r}}}$$

Apply the power rule $$$\int r^{n}\, dr = \frac{r^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$e {\color{red}{\int{r^{3} d r}}}=e {\color{red}{\frac{r^{1 + 3}}{1 + 3}}}=e {\color{red}{\left(\frac{r^{4}}{4}\right)}}$$

Therefore,

$$\int{e r^{3} d r} = \frac{e r^{4}}{4}$$

Add the constant of integration:

$$\int{e r^{3} d r} = \frac{e r^{4}}{4}+C$$

$$$\int e r^{3}\, dr = \frac{e r^{4}}{4} + C$$$A