Integral de $$$1 + \frac{2}{x}$$$

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

Integre termo a termo:

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

$$\int{\frac{2}{x} d x} + {\color{red}{\int{1 d x}}} = \int{\frac{2}{x} d x} + {\color{red}{x}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=2$$$ e $$$f{\left(x \right)} = \frac{1}{x}$$$:

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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