Integral de $$$1250 - 25 x$$$

Halla $$$\int \left(1250 - 25 x\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(1250 - 25 x\right)d x}}} = {\color{red}{\left(\int{1250 d x} - \int{25 x d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1250$$$:

$$- \int{25 x d x} + {\color{red}{\int{1250 d x}}} = - \int{25 x d x} + {\color{red}{\left(1250 x\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=25$$$ y $$$f{\left(x \right)} = x$$$:

$$1250 x - {\color{red}{\int{25 x d x}}} = 1250 x - {\color{red}{\left(25 \int{x d x}\right)}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=1$$$:

$$1250 x - 25 {\color{red}{\int{x d x}}}=1250 x - 25 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=1250 x - 25 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Por lo tanto,

$$\int{\left(1250 - 25 x\right)d x} = - \frac{25 x^{2}}{2} + 1250 x$$

Simplificar:

$$\int{\left(1250 - 25 x\right)d x} = \frac{25 x \left(100 - x\right)}{2}$$

Añade la constante de integración:

$$\int{\left(1250 - 25 x\right)d x} = \frac{25 x \left(100 - x\right)}{2}+C$$

$$$\int \left(1250 - 25 x\right)\, dx = \frac{25 x \left(100 - x\right)}{2} + C$$$A