Integral of $$$- 8 x^{4}$$$

Find $$$\int \left(- 8 x^{4}\right)\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-8$$$ and $$$f{\left(x \right)} = x^{4}$$$:

$${\color{red}{\int{\left(- 8 x^{4}\right)d x}}} = {\color{red}{\left(- 8 \int{x^{4} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$- 8 {\color{red}{\int{x^{4} d x}}}=- 8 {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- 8 {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

Therefore,

$$\int{\left(- 8 x^{4}\right)d x} = - \frac{8 x^{5}}{5}$$

Add the constant of integration:

$$\int{\left(- 8 x^{4}\right)d x} = - \frac{8 x^{5}}{5}+C$$

$$$\int \left(- 8 x^{4}\right)\, dx = - \frac{8 x^{5}}{5} + C$$$A