Integral de $$$\frac{5}{1 - x^{2}}$$$

Encontre $$$\int \frac{5}{1 - x^{2}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=5$$$ e $$$f{\left(x \right)} = \frac{1}{1 - x^{2}}$$$:

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

Efetue a decomposição em frações parciais (os passos podem ser vistos »):

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

Integre termo a termo:

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

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

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

Seja $$$u=x + 1$$$.

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

Portanto,

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

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

$$- 5 \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{5 {\color{red}{\int{\frac{1}{u} d u}}}}{2} = - 5 \int{\frac{1}{2 \left(x - 1\right)} d x} + \frac{5 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

Recorde que $$$u=x + 1$$$:

$$\frac{5 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} - 5 \int{\frac{1}{2 \left(x - 1\right)} d x} = \frac{5 \ln{\left(\left|{{\color{red}{\left(x + 1\right)}}}\right| \right)}}{2} - 5 \int{\frac{1}{2 \left(x - 1\right)} d x}$$

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

$$\frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - 5 {\color{red}{\int{\frac{1}{2 \left(x - 1\right)} d x}}} = \frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - 5 {\color{red}{\left(\frac{\int{\frac{1}{x - 1} d x}}{2}\right)}}$$

Seja $$$u=x - 1$$$.

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

A integral pode ser reescrita como

$$\frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 {\color{red}{\int{\frac{1}{x - 1} d x}}}}{2} = \frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 {\color{red}{\int{\frac{1}{u} d u}}}}{2}$$

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

$$\frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 {\color{red}{\int{\frac{1}{u} d u}}}}{2} = \frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

Recorde que $$$u=x - 1$$$:

$$\frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} = \frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2} - \frac{5 \ln{\left(\left|{{\color{red}{\left(x - 1\right)}}}\right| \right)}}{2}$$

Portanto,

$$\int{\frac{5}{1 - x^{2}} d x} = - \frac{5 \ln{\left(\left|{x - 1}\right| \right)}}{2} + \frac{5 \ln{\left(\left|{x + 1}\right| \right)}}{2}$$

Simplifique:

$$\int{\frac{5}{1 - x^{2}} d x} = \frac{5 \left(- \ln{\left(\left|{x - 1}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)}\right)}{2}$$

Adicione a constante de integração:

$$\int{\frac{5}{1 - x^{2}} d x} = \frac{5 \left(- \ln{\left(\left|{x - 1}\right| \right)} + \ln{\left(\left|{x + 1}\right| \right)}\right)}{2}+C$$

$$$\int \frac{5}{1 - x^{2}}\, dx = \frac{5 \left(- \ln\left(\left|{x - 1}\right|\right) + \ln\left(\left|{x + 1}\right|\right)\right)}{2} + C$$$A