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

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

Zij $$$u=x y$$$.

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

De integraal kan worden herschreven als

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

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

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

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

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

We herinneren eraan dat $$$u=x y$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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