Integral de $$$\frac{1}{2 x^{3}}$$$

Encontre $$$\int \frac{1}{2 x^{3}}\, dx$$$.

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 $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-3$$$:

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

Portanto,

$$\int{\frac{1}{2 x^{3}} d x} = - \frac{1}{4 x^{2}}$$

Adicione a constante de integração:

$$\int{\frac{1}{2 x^{3}} d x} = - \frac{1}{4 x^{2}}+C$$

$$$\int \frac{1}{2 x^{3}}\, dx = - \frac{1}{4 x^{2}} + C$$$A