$$$e \sin{\left(x \right)}$$$ 的积分

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

对 $$$c=e$$$ 和 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

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

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

因此，

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

加上积分常数：

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

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