Integral de $$$\frac{1}{2 a^{6} x^{6}}$$$ em relação a $$$x$$$

Encontre $$$\int \frac{1}{2 a^{6} x^{6}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{2 a^{6}}$$$ e $$$f{\left(x \right)} = \frac{1}{x^{6}}$$$:

$${\color{red}{\int{\frac{1}{2 a^{6} x^{6}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{6}} d x}}{2 a^{6}}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-6$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{6}} d x}}}}{2 a^{6}}=\frac{{\color{red}{\int{x^{-6} d x}}}}{2 a^{6}}=\frac{{\color{red}{\frac{x^{-6 + 1}}{-6 + 1}}}}{2 a^{6}}=\frac{{\color{red}{\left(- \frac{x^{-5}}{5}\right)}}}{2 a^{6}}=\frac{{\color{red}{\left(- \frac{1}{5 x^{5}}\right)}}}{2 a^{6}}$$

Portanto,

$$\int{\frac{1}{2 a^{6} x^{6}} d x} = - \frac{1}{10 a^{6} x^{5}}$$

Adicione a constante de integração:

$$\int{\frac{1}{2 a^{6} x^{6}} d x} = - \frac{1}{10 a^{6} x^{5}}+C$$

$$$\int \frac{1}{2 a^{6} x^{6}}\, dx = - \frac{1}{10 a^{6} x^{5}} + C$$$A