Integral of $$$\frac{x}{e^{3}}$$$

Find $$$\int \frac{x}{e^{3}}\, dx$$$.

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

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

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

$$\frac{{\color{red}{\int{x d x}}}}{e^{3}}=\frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{e^{3}}=\frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{e^{3}}$$

Therefore,

$$\int{\frac{x}{e^{3}} d x} = \frac{x^{2}}{2 e^{3}}$$

Add the constant of integration:

$$\int{\frac{x}{e^{3}} d x} = \frac{x^{2}}{2 e^{3}}+C$$

$$$\int \frac{x}{e^{3}}\, dx = \frac{x^{2}}{2 e^{3}} + C$$$A