Integral de $$$- 2 x + 10 e$$$

Encontre $$$\int \left(- 2 x + 10 e\right)\, dx$$$.

Integre termo a termo:

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

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

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

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

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=10 e$$$:

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

Portanto,

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

Simplifique:

$$\int{\left(- 2 x + 10 e\right)d x} = x \left(- x + 10 e\right)$$

Adicione a constante de integração:

$$\int{\left(- 2 x + 10 e\right)d x} = x \left(- x + 10 e\right)+C$$

$$$\int \left(- 2 x + 10 e\right)\, dx = x \left(- x + 10 e\right) + C$$$A