Integral de $$$\frac{x}{e^{5}}$$$

Encontre $$$\int \frac{x}{e^{5}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=e^{-5}$$$ e $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{x}{e^{5}} d x}}} = {\color{red}{\frac{\int{x d x}}{e^{5}}}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$\frac{{\color{red}{\int{x d x}}}}{e^{5}}=\frac{{\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{e^{5}}=\frac{{\color{red}{\left(\frac{x^{2}}{2}\right)}}}{e^{5}}$$

Portanto,

$$\int{\frac{x}{e^{5}} d x} = \frac{x^{2}}{2 e^{5}}$$

Adicione a constante de integração:

$$\int{\frac{x}{e^{5}} d x} = \frac{x^{2}}{2 e^{5}}+C$$

$$$\int \frac{x}{e^{5}}\, dx = \frac{x^{2}}{2 e^{5}} + C$$$A