Integral de $$$350 - 400 x$$$

Encontre $$$\int \left(350 - 400 x\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(350 - 400 x\right)d x}}} = {\color{red}{\left(\int{350 d x} - \int{400 x d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=350$$$:

$$- \int{400 x d x} + {\color{red}{\int{350 d x}}} = - \int{400 x d x} + {\color{red}{\left(350 x\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=400$$$ e $$$f{\left(x \right)} = x$$$:

$$350 x - {\color{red}{\int{400 x d x}}} = 350 x - {\color{red}{\left(400 \int{x d x}\right)}}$$

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

$$350 x - 400 {\color{red}{\int{x d x}}}=350 x - 400 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=350 x - 400 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Portanto,

$$\int{\left(350 - 400 x\right)d x} = - 200 x^{2} + 350 x$$

Simplifique:

$$\int{\left(350 - 400 x\right)d x} = 50 x \left(7 - 4 x\right)$$

Adicione a constante de integração:

$$\int{\left(350 - 400 x\right)d x} = 50 x \left(7 - 4 x\right)+C$$

$$$\int \left(350 - 400 x\right)\, dx = 50 x \left(7 - 4 x\right) + C$$$A