$$$\sin{\left(x \right)} + e$$$ 的积分

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=e$$$：

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

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

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

因此，

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

加上积分常数：

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

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