Integral of $$$x_{0}^{4} y_{0}^{4}$$$ with respect to $$$x_{0}$$$

Find $$$\int x_{0}^{4} y_{0}^{4}\, dx_{0}$$$.

Apply the constant multiple rule $$$\int c f{\left(x_{0} \right)}\, dx_{0} = c \int f{\left(x_{0} \right)}\, dx_{0}$$$ with $$$c=y_{0}^{4}$$$ and $$$f{\left(x_{0} \right)} = x_{0}^{4}$$$:

$${\color{red}{\int{x_{0}^{4} y_{0}^{4} d x_{0}}}} = {\color{red}{y_{0}^{4} \int{x_{0}^{4} d x_{0}}}}$$

Apply the power rule $$$\int x_{0}^{n}\, dx_{0} = \frac{x_{0}^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$y_{0}^{4} {\color{red}{\int{x_{0}^{4} d x_{0}}}}=y_{0}^{4} {\color{red}{\frac{x_{0}^{1 + 4}}{1 + 4}}}=y_{0}^{4} {\color{red}{\left(\frac{x_{0}^{5}}{5}\right)}}$$

Therefore,

$$\int{x_{0}^{4} y_{0}^{4} d x_{0}} = \frac{x_{0}^{5} y_{0}^{4}}{5}$$

Add the constant of integration:

$$\int{x_{0}^{4} y_{0}^{4} d x_{0}} = \frac{x_{0}^{5} y_{0}^{4}}{5}+C$$

$$$\int x_{0}^{4} y_{0}^{4}\, dx_{0} = \frac{x_{0}^{5} y_{0}^{4}}{5} + C$$$A