$$$\cos{\left(x \right)} + 1$$$ 的積分

求$$$\int \left(\cos{\left(x \right)} + 1\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(\cos{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} + \int{\cos{\left(x \right)} d x}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$\int{\cos{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = \int{\cos{\left(x \right)} d x} + {\color{red}{x}}$$

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

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

因此，

$$\int{\left(\cos{\left(x \right)} + 1\right)d x} = x + \sin{\left(x \right)}$$

加上積分常數：

$$\int{\left(\cos{\left(x \right)} + 1\right)d x} = x + \sin{\left(x \right)}+C$$

$$$\int \left(\cos{\left(x \right)} + 1\right)\, dx = \left(x + \sin{\left(x \right)}\right) + C$$$A