Integral de $$$x y e^{2}$$$ em relação a $$$x$$$

Encontre $$$\int x y e^{2}\, dx$$$.

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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