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

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

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

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-6$$$:

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

Therefore,

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

Add the constant of integration:

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

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