Integraal van $$$y \sin{\left(2 x \right)}$$$ met betrekking tot $$$x$$$

Bepaal $$$\int y \sin{\left(2 x \right)}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=y$$$ en $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$:

$${\color{red}{\int{y \sin{\left(2 x \right)} d x}}} = {\color{red}{y \int{\sin{\left(2 x \right)} d x}}}$$

Zij $$$u=2 x$$$.

Dan $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{du}{2}$$$.

Dus,

$$y {\color{red}{\int{\sin{\left(2 x \right)} d x}}} = y {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{1}{2}$$$ en $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$y {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}} = y {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}$$

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

$$\frac{y {\color{red}{\int{\sin{\left(u \right)} d u}}}}{2} = \frac{y {\color{red}{\left(- \cos{\left(u \right)}\right)}}}{2}$$

We herinneren eraan dat $$$u=2 x$$$:

$$- \frac{y \cos{\left({\color{red}{u}} \right)}}{2} = - \frac{y \cos{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

Dus,

$$\int{y \sin{\left(2 x \right)} d x} = - \frac{y \cos{\left(2 x \right)}}{2}$$

Voeg de integratieconstante toe:

$$\int{y \sin{\left(2 x \right)} d x} = - \frac{y \cos{\left(2 x \right)}}{2}+C$$

$$$\int y \sin{\left(2 x \right)}\, dx = - \frac{y \cos{\left(2 x \right)}}{2} + C$$$A