$$$\cos{\left(\frac{u}{v} \right)}$$$ 對 $$$u$$$ 的積分

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

令 $$$w=\frac{u}{v}$$$。

則 $$$dw=\left(\frac{u}{v}\right)^{\prime }du = \frac{du}{v}$$$ (步驟見»)，並可得 $$$du = v dw$$$。

該積分變為

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

套用常數倍法則 $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$，使用 $$$c=v$$$ 與 $$$f{\left(w \right)} = \cos{\left(w \right)}$$$：

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

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

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

回顧一下 $$$w=\frac{u}{v}$$$：

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

因此，

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

加上積分常數：

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

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