Integraal van $$$- \cos{\left(\frac{x}{y} \right)}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int \left(- \cos{\left(\frac{x}{y} \right)}\right)\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=-1$$$ en $$$f{\left(x \right)} = \cos{\left(\frac{x}{y} \right)}$$$:

$${\color{red}{\int{\left(- \cos{\left(\frac{x}{y} \right)}\right)d x}}} = {\color{red}{\left(- \int{\cos{\left(\frac{x}{y} \right)} d x}\right)}}$$

Zij $$$u=\frac{x}{y}$$$.

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

Dus,

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

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=y$$$ en $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

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

We herinneren eraan dat $$$u=\frac{x}{y}$$$:

$$- y \sin{\left({\color{red}{u}} \right)} = - y \sin{\left({\color{red}{\frac{x}{y}}} \right)}$$

Dus,

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

Voeg de integratieconstante toe:

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

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