Integral of $$$\cos{\left(\frac{u}{v} \right)}$$$ with respect to $$$u$$$

Find $$$\int \cos{\left(\frac{u}{v} \right)}\, du$$$.

Let $$$w=\frac{u}{v}$$$.

Then $$$dw=\left(\frac{u}{v}\right)^{\prime }du = \frac{du}{v}$$$ (steps can be seen »), and we have that $$$du = v dw$$$.

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ with $$$c=v$$$ and $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

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

The integral of the cosine is $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

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

Recall that $$$w=\frac{u}{v}$$$:

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

Therefore,

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

Add the constant of integration:

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

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