$$$5 y^{2} \cos{\left(x \right)}$$$ 关于$$$x$$$的积分

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

对 $$$c=5 y^{2}$$$ 和 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

余弦函数的积分为 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：

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

因此，

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

加上积分常数：

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

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