$$$\pi \sin{\left(x \right)}$$$ 的積分

求$$$\int \pi \sin{\left(x \right)}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\pi$$$ 與 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$：

$${\color{red}{\int{\pi \sin{\left(x \right)} d x}}} = {\color{red}{\pi \int{\sin{\left(x \right)} d x}}}$$

正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：

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

因此，

$$\int{\pi \sin{\left(x \right)} d x} = - \pi \cos{\left(x \right)}$$

加上積分常數：

$$\int{\pi \sin{\left(x \right)} d x} = - \pi \cos{\left(x \right)}+C$$

$$$\int \pi \sin{\left(x \right)}\, dx = - \pi \cos{\left(x \right)} + C$$$A