Integral de $$$12 x - 12$$$

Halla $$$\int \left(12 x - 12\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(12 x - 12\right)d x}}} = {\color{red}{\left(- \int{12 d x} + \int{12 x d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=12$$$:

$$\int{12 x d x} - {\color{red}{\int{12 d x}}} = \int{12 x d x} - {\color{red}{\left(12 x\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=12$$$ y $$$f{\left(x \right)} = x$$$:

$$- 12 x + {\color{red}{\int{12 x d x}}} = - 12 x + {\color{red}{\left(12 \int{x d x}\right)}}$$

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

$$- 12 x + 12 {\color{red}{\int{x d x}}}=- 12 x + 12 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 12 x + 12 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Por lo tanto,

$$\int{\left(12 x - 12\right)d x} = 6 x^{2} - 12 x$$

Simplificar:

$$\int{\left(12 x - 12\right)d x} = 6 x \left(x - 2\right)$$

Añade la constante de integración:

$$\int{\left(12 x - 12\right)d x} = 6 x \left(x - 2\right)+C$$

$$$\int \left(12 x - 12\right)\, dx = 6 x \left(x - 2\right) + C$$$A