Integral de $$$x y^{2}$$$ con respecto a $$$x$$$

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=y^{2}$$$ y $$$f{\left(x \right)} = x$$$:

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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