Integral de $$$\frac{x - 5}{3 x - 2}$$$

Encontre $$$\int \frac{x - 5}{3 x - 2}\, dx$$$.

Reescreva o numerador do integrando como $$$x - 5=\frac{1}{3}\left(3 x - 2\right)- \frac{13}{3}$$$ e decomponha a fração:

$${\color{red}{\int{\frac{x - 5}{3 x - 2} d x}}} = {\color{red}{\int{\left(\frac{1}{3} - \frac{13}{3 \left(3 x - 2\right)}\right)d x}}}$$

Integre termo a termo:

$${\color{red}{\int{\left(\frac{1}{3} - \frac{13}{3 \left(3 x - 2\right)}\right)d x}}} = {\color{red}{\left(\int{\frac{1}{3} d x} - \int{\frac{13}{3 \left(3 x - 2\right)} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=\frac{1}{3}$$$:

$$- \int{\frac{13}{3 \left(3 x - 2\right)} d x} + {\color{red}{\int{\frac{1}{3} d x}}} = - \int{\frac{13}{3 \left(3 x - 2\right)} d x} + {\color{red}{\left(\frac{x}{3}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{13}{3}$$$ e $$$f{\left(x \right)} = \frac{1}{3 x - 2}$$$:

$$\frac{x}{3} - {\color{red}{\int{\frac{13}{3 \left(3 x - 2\right)} d x}}} = \frac{x}{3} - {\color{red}{\left(\frac{13 \int{\frac{1}{3 x - 2} d x}}{3}\right)}}$$

Seja $$$u=3 x - 2$$$.

Então $$$du=\left(3 x - 2\right)^{\prime }dx = 3 dx$$$ (veja os passos »), e obtemos $$$dx = \frac{du}{3}$$$.

Portanto,

$$\frac{x}{3} - \frac{13 {\color{red}{\int{\frac{1}{3 x - 2} d x}}}}{3} = \frac{x}{3} - \frac{13 {\color{red}{\int{\frac{1}{3 u} d u}}}}{3}$$

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

$$\frac{x}{3} - \frac{13 {\color{red}{\int{\frac{1}{3 u} d u}}}}{3} = \frac{x}{3} - \frac{13 {\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{3}\right)}}}{3}$$

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

$$\frac{x}{3} - \frac{13 {\color{red}{\int{\frac{1}{u} d u}}}}{9} = \frac{x}{3} - \frac{13 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{9}$$

Recorde que $$$u=3 x - 2$$$:

$$\frac{x}{3} - \frac{13 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{9} = \frac{x}{3} - \frac{13 \ln{\left(\left|{{\color{red}{\left(3 x - 2\right)}}}\right| \right)}}{9}$$

Portanto,

$$\int{\frac{x - 5}{3 x - 2} d x} = \frac{x}{3} - \frac{13 \ln{\left(\left|{3 x - 2}\right| \right)}}{9}$$

Adicione a constante de integração:

$$\int{\frac{x - 5}{3 x - 2} d x} = \frac{x}{3} - \frac{13 \ln{\left(\left|{3 x - 2}\right| \right)}}{9}+C$$

$$$\int \frac{x - 5}{3 x - 2}\, dx = \left(\frac{x}{3} - \frac{13 \ln\left(\left|{3 x - 2}\right|\right)}{9}\right) + C$$$A