Integral de $$$1 - 10 x^{2}$$$

Encontre $$$\int \left(1 - 10 x^{2}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(1 - 10 x^{2}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{10 x^{2} d x}\right)}}$$

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

$$- \int{10 x^{2} d x} + {\color{red}{\int{1 d x}}} = - \int{10 x^{2} d x} + {\color{red}{x}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=10$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$$x - {\color{red}{\int{10 x^{2} d x}}} = x - {\color{red}{\left(10 \int{x^{2} d x}\right)}}$$

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

$$x - 10 {\color{red}{\int{x^{2} d x}}}=x - 10 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=x - 10 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Portanto,

$$\int{\left(1 - 10 x^{2}\right)d x} = - \frac{10 x^{3}}{3} + x$$

Adicione a constante de integração:

$$\int{\left(1 - 10 x^{2}\right)d x} = - \frac{10 x^{3}}{3} + x+C$$

$$$\int \left(1 - 10 x^{2}\right)\, dx = \left(- \frac{10 x^{3}}{3} + x\right) + C$$$A