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

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

Seja $$$u=x^{2} - 1$$$.

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

Portanto,

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

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

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

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

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

Recorde que $$$u=x^{2} - 1$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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