Integral of $$$\frac{s}{e^{4}}$$$

Find $$$\int \frac{s}{e^{4}}\, ds$$$.

Apply the constant multiple rule $$$\int c f{\left(s \right)}\, ds = c \int f{\left(s \right)}\, ds$$$ with $$$c=e^{-4}$$$ and $$$f{\left(s \right)} = s$$$:

$${\color{red}{\int{\frac{s}{e^{4}} d s}}} = {\color{red}{\frac{\int{s d s}}{e^{4}}}}$$

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

$$\frac{{\color{red}{\int{s d s}}}}{e^{4}}=\frac{{\color{red}{\frac{s^{1 + 1}}{1 + 1}}}}{e^{4}}=\frac{{\color{red}{\left(\frac{s^{2}}{2}\right)}}}{e^{4}}$$

Therefore,

$$\int{\frac{s}{e^{4}} d s} = \frac{s^{2}}{2 e^{4}}$$

Add the constant of integration:

$$\int{\frac{s}{e^{4}} d s} = \frac{s^{2}}{2 e^{4}}+C$$

$$$\int \frac{s}{e^{4}}\, ds = \frac{s^{2}}{2 e^{4}} + C$$$A