Integral of $$$\frac{1}{2 a^{6} x^{5}}$$$ with respect to $$$x$$$

Find $$$\int \frac{1}{2 a^{6} x^{5}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2 a^{6}}$$$ and $$$f{\left(x \right)} = \frac{1}{x^{5}}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-5$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{5}} d x}}}}{2 a^{6}}=\frac{{\color{red}{\int{x^{-5} d x}}}}{2 a^{6}}=\frac{{\color{red}{\frac{x^{-5 + 1}}{-5 + 1}}}}{2 a^{6}}=\frac{{\color{red}{\left(- \frac{x^{-4}}{4}\right)}}}{2 a^{6}}=\frac{{\color{red}{\left(- \frac{1}{4 x^{4}}\right)}}}{2 a^{6}}$$

Therefore,

$$\int{\frac{1}{2 a^{6} x^{5}} d x} = - \frac{1}{8 a^{6} x^{4}}$$

Add the constant of integration:

$$\int{\frac{1}{2 a^{6} x^{5}} d x} = - \frac{1}{8 a^{6} x^{4}}+C$$

$$$\int \frac{1}{2 a^{6} x^{5}}\, dx = - \frac{1}{8 a^{6} x^{4}} + C$$$A