$$$z - 10 \sin{\left(x \right)}$$$ 對 $$$x$$$ 的積分

求$$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(z - 10 \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{z d x} - \int{10 \sin{\left(x \right)} d x}\right)}}$$

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

$$- \int{10 \sin{\left(x \right)} d x} + {\color{red}{\int{z d x}}} = - \int{10 \sin{\left(x \right)} d x} + {\color{red}{x z}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=10$$$ 與 $$$f{\left(x \right)} = \sin{\left(x \right)}$$$：

$$x z - {\color{red}{\int{10 \sin{\left(x \right)} d x}}} = x z - {\color{red}{\left(10 \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$x z - 10 {\color{red}{\int{\sin{\left(x \right)} d x}}} = x z - 10 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

因此，

$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}$$

加上積分常數：

$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}+C$$

$$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx = \left(x z + 10 \cos{\left(x \right)}\right) + C$$$A