Integral de $$$\frac{2}{x - 2}$$$

Encontre $$$\int \frac{2}{x - 2}\, dx$$$.

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

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

Seja $$$u=x - 2$$$.

Então $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (veja os passos »), e obtemos $$$dx = du$$$.

Portanto,

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

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

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

Recorde que $$$u=x - 2$$$:

$$2 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 2 \ln{\left(\left|{{\color{red}{\left(x - 2\right)}}}\right| \right)}$$

Portanto,

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

Adicione a constante de integração:

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

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