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

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

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

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

De integraal van de sinus is $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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