$$$e \sin{\left(x \right)}$$$ 的積分

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

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

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

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

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

因此，

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

加上積分常數：

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

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