Integral de $$$1 - 112 x^{3}$$$

Halla $$$\int \left(1 - 112 x^{3}\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(1 - 112 x^{3}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{112 x^{3} d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=1$$$:

$$- \int{112 x^{3} d x} + {\color{red}{\int{1 d x}}} = - \int{112 x^{3} d x} + {\color{red}{x}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=112$$$ y $$$f{\left(x \right)} = x^{3}$$$:

$$x - {\color{red}{\int{112 x^{3} d x}}} = x - {\color{red}{\left(112 \int{x^{3} d x}\right)}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=3$$$:

$$x - 112 {\color{red}{\int{x^{3} d x}}}=x - 112 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=x - 112 {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Por lo tanto,

$$\int{\left(1 - 112 x^{3}\right)d x} = - 28 x^{4} + x$$

Añade la constante de integración:

$$\int{\left(1 - 112 x^{3}\right)d x} = - 28 x^{4} + x+C$$

$$$\int \left(1 - 112 x^{3}\right)\, dx = \left(- 28 x^{4} + x\right) + C$$$A