Integral de $$$y^{2} - 1$$$

Halla $$$\int \left(y^{2} - 1\right)\, dy$$$.

Integra término a término:

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

Aplica la regla de la constante $$$\int c\, dy = c y$$$ con $$$c=1$$$:

$$\int{y^{2} d y} - {\color{red}{\int{1 d y}}} = \int{y^{2} d y} - {\color{red}{y}}$$

Aplica la regla de la potencia $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

$$$\int \left(y^{2} - 1\right)\, dy = \left(\frac{y^{3}}{3} - y\right) + C$$$A