$$$\cos{\left(\frac{u}{v} \right)}$$$ 关于$$$u$$$的积分

求$$$\int \cos{\left(\frac{u}{v} \right)}\, du$$$。

设$$$w=\frac{u}{v}$$$。

则$$$dw=\left(\frac{u}{v}\right)^{\prime }du = \frac{du}{v}$$$ (步骤见»)，并有$$$du = v dw$$$。

因此，

$${\color{red}{\int{\cos{\left(\frac{u}{v} \right)} d u}}} = {\color{red}{\int{v \cos{\left(w \right)} d w}}}$$

对 $$$c=v$$$ 和 $$$f{\left(w \right)} = \cos{\left(w \right)}$$$ 应用常数倍法则 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$：

$${\color{red}{\int{v \cos{\left(w \right)} d w}}} = {\color{red}{v \int{\cos{\left(w \right)} d w}}}$$

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

$$v {\color{red}{\int{\cos{\left(w \right)} d w}}} = v {\color{red}{\sin{\left(w \right)}}}$$

回忆一下 $$$w=\frac{u}{v}$$$:

$$v \sin{\left({\color{red}{w}} \right)} = v \sin{\left({\color{red}{\frac{u}{v}}} \right)}$$

因此，

$$\int{\cos{\left(\frac{u}{v} \right)} d u} = v \sin{\left(\frac{u}{v} \right)}$$

加上积分常数：

$$\int{\cos{\left(\frac{u}{v} \right)} d u} = v \sin{\left(\frac{u}{v} \right)}+C$$

$$$\int \cos{\left(\frac{u}{v} \right)}\, du = v \sin{\left(\frac{u}{v} \right)} + C$$$A