Integral de $$$\frac{x^{2}}{7 - x^{3}}$$$

Encontre $$$\int \frac{x^{2}}{7 - x^{3}}\, dx$$$.

Seja $$$u=7 - x^{3}$$$.

Então $$$du=\left(7 - x^{3}\right)^{\prime }dx = - 3 x^{2} dx$$$ (veja os passos »), e obtemos $$$x^{2} dx = - \frac{du}{3}$$$.

Logo,

$${\color{red}{\int{\frac{x^{2}}{7 - x^{3}} d x}}} = {\color{red}{\int{\left(- \frac{1}{3 u}\right)d u}}}$$

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

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

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

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

Recorde que $$$u=7 - x^{3}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{3} = - \frac{\ln{\left(\left|{{\color{red}{\left(7 - x^{3}\right)}}}\right| \right)}}{3}$$

Portanto,

$$\int{\frac{x^{2}}{7 - x^{3}} d x} = - \frac{\ln{\left(\left|{x^{3} - 7}\right| \right)}}{3}$$

Adicione a constante de integração:

$$\int{\frac{x^{2}}{7 - x^{3}} d x} = - \frac{\ln{\left(\left|{x^{3} - 7}\right| \right)}}{3}+C$$

$$$\int \frac{x^{2}}{7 - x^{3}}\, dx = - \frac{\ln\left(\left|{x^{3} - 7}\right|\right)}{3} + C$$$A