Integral of $$$8 x^{7}$$$

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

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

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

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

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

Therefore,

$$\int{8 x^{7} d x} = x^{8}$$

Add the constant of integration:

$$\int{8 x^{7} d x} = x^{8}+C$$

$$$\int 8 x^{7}\, dx = x^{8} + C$$$A