Integral of $$$\frac{6}{a^{8} x^{7}}$$$ with respect to $$$x$$$

Find $$$\int \frac{6}{a^{8} x^{7}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{6}{a^{8}}$$$ and $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$:

$${\color{red}{\int{\frac{6}{a^{8} x^{7}} d x}}} = {\color{red}{\left(\frac{6 \int{\frac{1}{x^{7}} d x}}{a^{8}}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-7$$$:

$$\frac{6 {\color{red}{\int{\frac{1}{x^{7}} d x}}}}{a^{8}}=\frac{6 {\color{red}{\int{x^{-7} d x}}}}{a^{8}}=\frac{6 {\color{red}{\frac{x^{-7 + 1}}{-7 + 1}}}}{a^{8}}=\frac{6 {\color{red}{\left(- \frac{x^{-6}}{6}\right)}}}{a^{8}}=\frac{6 {\color{red}{\left(- \frac{1}{6 x^{6}}\right)}}}{a^{8}}$$

Therefore,

$$\int{\frac{6}{a^{8} x^{7}} d x} = - \frac{1}{a^{8} x^{6}}$$

Add the constant of integration:

$$\int{\frac{6}{a^{8} x^{7}} d x} = - \frac{1}{a^{8} x^{6}}+C$$

$$$\int \frac{6}{a^{8} x^{7}}\, dx = - \frac{1}{a^{8} x^{6}} + C$$$A