Integral de $$$- a^{2} - y^{2} + 1$$$ em relação a $$$y$$$

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dy = c y$$$ usando $$$c=1$$$:

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

Aplique a regra da constante $$$\int c\, dy = c y$$$ usando $$$c=a^{2}$$$:

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

Aplique a regra da potência $$$\int y^{n}\, dy = \frac{y^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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