Integral de $$$- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}$$$

Halla $$$\int \left(- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}\right)\, dx$$$.

La entrada se reescribe: $$$\int{\left(- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}\right)d x}=\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x}$$$.

Simplificar el integrando:

$${\color{red}{\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x}}} = {\color{red}{\int{\left(\frac{1}{3} - x^{2}\right)d x}}}$$

Integra término a término:

$${\color{red}{\int{\left(\frac{1}{3} - x^{2}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{3} d x} - \int{x^{2} d x}\right)}}$$

Aplica la regla de la constante $$$\int c\, dx = c x$$$ con $$$c=\frac{1}{3}$$$:

$$- \int{x^{2} d x} + {\color{red}{\int{\frac{1}{3} d x}}} = - \int{x^{2} d x} + {\color{red}{\left(\frac{x}{3}\right)}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=2$$$:

$$\frac{x}{3} - {\color{red}{\int{x^{2} d x}}}=\frac{x}{3} - {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{x}{3} - {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Por lo tanto,

$$\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x} = - \frac{x^{3}}{3} + \frac{x}{3}$$

Simplificar:

$$\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x} = \frac{x \left(1 - x^{2}\right)}{3}$$

Añade la constante de integración:

$$\int{x^{2} \left(-1 + \frac{1}{3 x^{2}}\right) d x} = \frac{x \left(1 - x^{2}\right)}{3}+C$$

$$$\int \left(- \frac{x^{2} \left(3 - \frac{1}{x^{2}}\right)}{3}\right)\, dx = \frac{x \left(1 - x^{2}\right)}{3} + C$$$A