$$$\sin{\left(\frac{x}{k} \right)}$$$ 关于$$$x$$$的积分

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

设$$$u=\frac{x}{k}$$$。

则$$$du=\left(\frac{x}{k}\right)^{\prime }dx = \frac{dx}{k}$$$ (步骤见»)，并有$$$dx = k du$$$。

积分变为

$${\color{red}{\int{\sin{\left(\frac{x}{k} \right)} d x}}} = {\color{red}{\int{k \sin{\left(u \right)} d u}}}$$

对 $$$c=k$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$${\color{red}{\int{k \sin{\left(u \right)} d u}}} = {\color{red}{k \int{\sin{\left(u \right)} d u}}}$$

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

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

回忆一下 $$$u=\frac{x}{k}$$$:

$$- k \cos{\left({\color{red}{u}} \right)} = - k \cos{\left({\color{red}{\frac{x}{k}}} \right)}$$

因此，

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

加上积分常数：

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

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