Integral de $$$x e^{4}$$$

Halla $$$\int x e^{4}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=e^{4}$$$ y $$$f{\left(x \right)} = x$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

$$$\int x e^{4}\, dx = \frac{x^{2} e^{4}}{2} + C$$$A