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

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

Integre termo a termo:

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

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

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

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=3$$$ e $$$f{\left(x \right)} = x^{2}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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