Integral de $$$x - \frac{1}{2}$$$

Halla $$$\int \left(x - \frac{1}{2}\right)\, dx$$$.

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=\frac{1}{2}$$$:

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

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

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

Por lo tanto,

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

Simplificar:

$$\int{\left(x - \frac{1}{2}\right)d x} = \frac{x \left(x - 1\right)}{2}$$

Añade la constante de integración:

$$\int{\left(x - \frac{1}{2}\right)d x} = \frac{x \left(x - 1\right)}{2}+C$$

$$$\int \left(x - \frac{1}{2}\right)\, dx = \frac{x \left(x - 1\right)}{2} + C$$$A