$$$x_{0}^{4} y_{0}^{4}$$$ の $$$x_{0}$$$ に関する積分

$$$\int x_{0}^{4} y_{0}^{4}\, dx_{0}$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x_{0} \right)}\, dx_{0} = c \int f{\left(x_{0} \right)}\, dx_{0}$$$ を、$$$c=y_{0}^{4}$$$ と $$$f{\left(x_{0} \right)} = x_{0}^{4}$$$ に対して適用する:

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

$$$n=4$$$ を用いて、べき乗の法則 $$$\int x_{0}^{n}\, dx_{0} = \frac{x_{0}^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

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

したがって、

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

積分定数を加える:

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

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