Integral de $$$- _1 x^{2} + 1$$$ em relação a $$$x$$$

Encontre $$$\int \left(- _1 x^{2} + 1\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(- _1 x^{2} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{_1 x^{2} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

$$- \int{_1 x^{2} d x} + {\color{red}{\int{1 d x}}} = - \int{_1 x^{2} d x} + {\color{red}{x}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=_1$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$$x - {\color{red}{\int{_1 x^{2} d x}}} = x - {\color{red}{_1 \int{x^{2} d x}}}$$

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

$$- _1 {\color{red}{\int{x^{2} d x}}} + x=- _1 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}} + x=- _1 {\color{red}{\left(\frac{x^{3}}{3}\right)}} + x$$

Portanto,

$$\int{\left(- _1 x^{2} + 1\right)d x} = - \frac{_1 x^{3}}{3} + x$$

Adicione a constante de integração:

$$\int{\left(- _1 x^{2} + 1\right)d x} = - \frac{_1 x^{3}}{3} + x+C$$

$$$\int \left(- _1 x^{2} + 1\right)\, dx = \left(- \frac{_1 x^{3}}{3} + x\right) + C$$$A