$$$5 y^{2} \cos{\left(x \right)}$$$ の $$$x$$$ に関する積分

$$$\int 5 y^{2} \cos{\left(x \right)}\, dx$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=5 y^{2}$$$ と $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用する:

$${\color{red}{\int{5 y^{2} \cos{\left(x \right)} d x}}} = {\color{red}{\left(5 y^{2} \int{\cos{\left(x \right)} d x}\right)}}$$

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

$$5 y^{2} {\color{red}{\int{\cos{\left(x \right)} d x}}} = 5 y^{2} {\color{red}{\sin{\left(x \right)}}}$$

したがって、

$$\int{5 y^{2} \cos{\left(x \right)} d x} = 5 y^{2} \sin{\left(x \right)}$$

積分定数を加える:

$$\int{5 y^{2} \cos{\left(x \right)} d x} = 5 y^{2} \sin{\left(x \right)}+C$$

$$$\int 5 y^{2} \cos{\left(x \right)}\, dx = 5 y^{2} \sin{\left(x \right)} + C$$$A