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

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

Integre termo a termo:

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

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

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

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

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

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

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

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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