Integral de $$$3 x - y$$$ con respecto a $$$x$$$

Halla $$$\int \left(3 x - y\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(3 x - y\right)d x}}} = {\color{red}{\left(\int{3 x d x} - \int{y d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=y$$$:

$$\int{3 x d x} - {\color{red}{\int{y d x}}} = \int{3 x d x} - {\color{red}{x y}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=3$$$ y $$$f{\left(x \right)} = x$$$:

$$- x y + {\color{red}{\int{3 x d x}}} = - x y + {\color{red}{\left(3 \int{x d x}\right)}}$$

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

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

Por lo tanto,

$$\int{\left(3 x - y\right)d x} = \frac{3 x^{2}}{2} - x y$$

Simplificar:

$$\int{\left(3 x - y\right)d x} = \frac{x \left(3 x - 2 y\right)}{2}$$

Añade la constante de integración:

$$\int{\left(3 x - y\right)d x} = \frac{x \left(3 x - 2 y\right)}{2}+C$$

$$$\int \left(3 x - y\right)\, dx = \frac{x \left(3 x - 2 y\right)}{2} + C$$$A