Integral de $$$11 - 12 x^{2}$$$

Encontre $$$\int \left(11 - 12 x^{2}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(11 - 12 x^{2}\right)d x}}} = {\color{red}{\left(\int{11 d x} - \int{12 x^{2} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=11$$$:

$$- \int{12 x^{2} d x} + {\color{red}{\int{11 d x}}} = - \int{12 x^{2} d x} + {\color{red}{\left(11 x\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=12$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$$11 x - {\color{red}{\int{12 x^{2} d x}}} = 11 x - {\color{red}{\left(12 \int{x^{2} d x}\right)}}$$

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

$$11 x - 12 {\color{red}{\int{x^{2} d x}}}=11 x - 12 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=11 x - 12 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Portanto,

$$\int{\left(11 - 12 x^{2}\right)d x} = - 4 x^{3} + 11 x$$

Simplifique:

$$\int{\left(11 - 12 x^{2}\right)d x} = x \left(11 - 4 x^{2}\right)$$

Adicione a constante de integração:

$$\int{\left(11 - 12 x^{2}\right)d x} = x \left(11 - 4 x^{2}\right)+C$$

$$$\int \left(11 - 12 x^{2}\right)\, dx = x \left(11 - 4 x^{2}\right) + C$$$A