$$$- 8 \cos{\left(t \right)} - 1$$$の積分

$$$\int \left(- 8 \cos{\left(t \right)} - 1\right)\, dt$$$ を求めよ。

項別に積分せよ:

$${\color{red}{\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t}}} = {\color{red}{\left(- \int{1 d t} - \int{8 \cos{\left(t \right)} d t}\right)}}$$

$$$c=1$$$ に対して定数則 $$$\int c\, dt = c t$$$ を適用する:

$$- \int{8 \cos{\left(t \right)} d t} - {\color{red}{\int{1 d t}}} = - \int{8 \cos{\left(t \right)} d t} - {\color{red}{t}}$$

定数倍の法則 $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ を、$$$c=8$$$ と $$$f{\left(t \right)} = \cos{\left(t \right)}$$$ に対して適用する:

$$- t - {\color{red}{\int{8 \cos{\left(t \right)} d t}}} = - t - {\color{red}{\left(8 \int{\cos{\left(t \right)} d t}\right)}}$$

余弦の積分は$$$\int{\cos{\left(t \right)} d t} = \sin{\left(t \right)}$$$:

$$- t - 8 {\color{red}{\int{\cos{\left(t \right)} d t}}} = - t - 8 {\color{red}{\sin{\left(t \right)}}}$$

したがって、

$$\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t} = - t - 8 \sin{\left(t \right)}$$

積分定数を加える:

$$\int{\left(- 8 \cos{\left(t \right)} - 1\right)d t} = - t - 8 \sin{\left(t \right)}+C$$

$$$\int \left(- 8 \cos{\left(t \right)} - 1\right)\, dt = \left(- t - 8 \sin{\left(t \right)}\right) + C$$$A