Integral de $$$- \frac{x^{3}}{e^{6}} + x e^{2}$$$

Encontre $$$\int \left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x}}} = {\color{red}{\left(\int{x e^{2} d x} - \int{\frac{x^{3}}{e^{6}} d x}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=e^{2}$$$ e $$$f{\left(x \right)} = x$$$:

$$- \int{\frac{x^{3}}{e^{6}} d x} + {\color{red}{\int{x e^{2} d x}}} = - \int{\frac{x^{3}}{e^{6}} d x} + {\color{red}{e^{2} \int{x d x}}}$$

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

$$- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\int{x d x}}}=- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{x^{3}}{e^{6}} d x} + e^{2} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=e^{-6}$$$ e $$$f{\left(x \right)} = x^{3}$$$:

$$\frac{x^{2} e^{2}}{2} - {\color{red}{\int{\frac{x^{3}}{e^{6}} d x}}} = \frac{x^{2} e^{2}}{2} - {\color{red}{\frac{\int{x^{3} d x}}{e^{6}}}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=3$$$:

$$\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\int{x^{3} d x}}}}{e^{6}}=\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\frac{x^{1 + 3}}{1 + 3}}}}{e^{6}}=\frac{x^{2} e^{2}}{2} - \frac{{\color{red}{\left(\frac{x^{4}}{4}\right)}}}{e^{6}}$$

Portanto,

$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = - \frac{x^{4}}{4 e^{6}} + \frac{x^{2} e^{2}}{2}$$

Simplifique:

$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}}$$

Adicione a constante de integração:

$$\int{\left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)d x} = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}}+C$$

$$$\int \left(- \frac{x^{3}}{e^{6}} + x e^{2}\right)\, dx = \frac{x^{2} \left(- x^{2} + 2 e^{8}\right)}{4 e^{6}} + C$$$A