Integraal van $$$\cos{\left(\frac{u}{v} \right)}$$$ met betrekking tot $$$u$$$

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

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

Dan $$$dw=\left(\frac{u}{v}\right)^{\prime }du = \frac{du}{v}$$$ (de stappen zijn te zien »), en dan geldt dat $$$du = v dw$$$.

De integraal kan worden herschreven als

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

Pas de constante-veelvoudregel $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ toe met $$$c=v$$$ en $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

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

De integraal van de cosinus is $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

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

We herinneren eraan dat $$$w=\frac{u}{v}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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