Integral de $$$\frac{3}{x - 4}$$$

Encontre $$$\int \frac{3}{x - 4}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=3$$$ e $$$f{\left(x \right)} = \frac{1}{x - 4}$$$:

$${\color{red}{\int{\frac{3}{x - 4} d x}}} = {\color{red}{\left(3 \int{\frac{1}{x - 4} d x}\right)}}$$

Seja $$$u=x - 4$$$.

Então $$$du=\left(x - 4\right)^{\prime }dx = 1 dx$$$ (veja os passos »), e obtemos $$$dx = du$$$.

Logo,

$$3 {\color{red}{\int{\frac{1}{x - 4} d x}}} = 3 {\color{red}{\int{\frac{1}{u} d u}}}$$

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

$$3 {\color{red}{\int{\frac{1}{u} d u}}} = 3 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Recorde que $$$u=x - 4$$$:

$$3 \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = 3 \ln{\left(\left|{{\color{red}{\left(x - 4\right)}}}\right| \right)}$$

Portanto,

$$\int{\frac{3}{x - 4} d x} = 3 \ln{\left(\left|{x - 4}\right| \right)}$$

Adicione a constante de integração:

$$\int{\frac{3}{x - 4} d x} = 3 \ln{\left(\left|{x - 4}\right| \right)}+C$$

$$$\int \frac{3}{x - 4}\, dx = 3 \ln\left(\left|{x - 4}\right|\right) + C$$$A