$$$8498000 - 212450 t$$$の積分

$$$\int \left(8498000 - 212450 t\right)\, dt$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(8498000 - 212450 t\right)d t}}} = {\color{red}{\left(\int{8498000 d t} - \int{212450 t d t}\right)}}$$

$$$c=8498000$$$ に対して定数則 $$$\int c\, dt = c t$$$ を適用する:

$$- \int{212450 t d t} + {\color{red}{\int{8498000 d t}}} = - \int{212450 t d t} + {\color{red}{\left(8498000 t\right)}}$$

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=212450$$$ と $$$f{\left(t \right)} = t$$$ に対して適用する:

$$8498000 t - {\color{red}{\int{212450 t d t}}} = 8498000 t - {\color{red}{\left(212450 \int{t d t}\right)}}$$

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

$$8498000 t - 212450 {\color{red}{\int{t d t}}}=8498000 t - 212450 {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}=8498000 t - 212450 {\color{red}{\left(\frac{t^{2}}{2}\right)}}$$

したがって、

$$\int{\left(8498000 - 212450 t\right)d t} = - 106225 t^{2} + 8498000 t$$

簡単化せよ:

$$\int{\left(8498000 - 212450 t\right)d t} = 106225 t \left(80 - t\right)$$

積分定数を加える:

$$\int{\left(8498000 - 212450 t\right)d t} = 106225 t \left(80 - t\right)+C$$

$$$\int \left(8498000 - 212450 t\right)\, dt = 106225 t \left(80 - t\right) + C$$$A