Integral de $$$x^{3} - 9 x - 9$$$

Halla $$$\int \left(x^{3} - 9 x - 9\right)\, dx$$$.

Integra término a término:

$${\color{red}{\int{\left(x^{3} - 9 x - 9\right)d x}}} = {\color{red}{\left(- \int{9 d x} - \int{9 x d x} + \int{x^{3} d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=9$$$:

$$- \int{9 x d x} + \int{x^{3} d x} - {\color{red}{\int{9 d x}}} = - \int{9 x d x} + \int{x^{3} d x} - {\color{red}{\left(9 x\right)}}$$

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

$$- 9 x - \int{9 x d x} + {\color{red}{\int{x^{3} d x}}}=- 9 x - \int{9 x d x} + {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=- 9 x - \int{9 x d x} + {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=9$$$ y $$$f{\left(x \right)} = x$$$:

$$\frac{x^{4}}{4} - 9 x - {\color{red}{\int{9 x d x}}} = \frac{x^{4}}{4} - 9 x - {\color{red}{\left(9 \int{x d x}\right)}}$$

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

$$\frac{x^{4}}{4} - 9 x - 9 {\color{red}{\int{x d x}}}=\frac{x^{4}}{4} - 9 x - 9 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{4}}{4} - 9 x - 9 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Por lo tanto,

$$\int{\left(x^{3} - 9 x - 9\right)d x} = \frac{x^{4}}{4} - \frac{9 x^{2}}{2} - 9 x$$

Simplificar:

$$\int{\left(x^{3} - 9 x - 9\right)d x} = \frac{x \left(x^{3} - 18 x - 36\right)}{4}$$

Añade la constante de integración:

$$\int{\left(x^{3} - 9 x - 9\right)d x} = \frac{x \left(x^{3} - 18 x - 36\right)}{4}+C$$

$$$\int \left(x^{3} - 9 x - 9\right)\, dx = \frac{x \left(x^{3} - 18 x - 36\right)}{4} + C$$$A