Integral of $$$p e^{2}$$$

Find $$$\int p e^{2}\, dp$$$.

Apply the constant multiple rule $$$\int c f{\left(p \right)}\, dp = c \int f{\left(p \right)}\, dp$$$ with $$$c=e^{2}$$$ and $$$f{\left(p \right)} = p$$$:

$${\color{red}{\int{p e^{2} d p}}} = {\color{red}{e^{2} \int{p d p}}}$$

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

$$e^{2} {\color{red}{\int{p d p}}}=e^{2} {\color{red}{\frac{p^{1 + 1}}{1 + 1}}}=e^{2} {\color{red}{\left(\frac{p^{2}}{2}\right)}}$$

Therefore,

$$\int{p e^{2} d p} = \frac{p^{2} e^{2}}{2}$$

Add the constant of integration:

$$\int{p e^{2} d p} = \frac{p^{2} e^{2}}{2}+C$$

$$$\int p e^{2}\, dp = \frac{p^{2} e^{2}}{2} + C$$$A