Integral de $$$- 3 x^{2} + \frac{1}{x}$$$

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

Integra término a término:

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

La integral de $$$\frac{1}{x}$$$ es $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=3$$$ y $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Por lo tanto,

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

Añade la constante de integración:

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

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