$$$\frac{1}{4} - 5 \cos{\left(x \right)}$$$の積分

$$$\int \left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)\, dx$$$ を求めよ。

項別に積分せよ:

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

$$$c=\frac{1}{4}$$$ に対して定数則 $$$\int c\, dx = c x$$$ を適用する:

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=5$$$ と $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用する:

$$\frac{x}{4} - {\color{red}{\int{5 \cos{\left(x \right)} d x}}} = \frac{x}{4} - {\color{red}{\left(5 \int{\cos{\left(x \right)} d x}\right)}}$$

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

$$\frac{x}{4} - 5 {\color{red}{\int{\cos{\left(x \right)} d x}}} = \frac{x}{4} - 5 {\color{red}{\sin{\left(x \right)}}}$$

したがって、

$$\int{\left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)d x} = \frac{x}{4} - 5 \sin{\left(x \right)}$$

積分定数を加える:

$$\int{\left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)d x} = \frac{x}{4} - 5 \sin{\left(x \right)}+C$$

$$$\int \left(\frac{1}{4} - 5 \cos{\left(x \right)}\right)\, dx = \left(\frac{x}{4} - 5 \sin{\left(x \right)}\right) + C$$$A