Integraal van $$$2 \sin{\left(2 t \right)}$$$

Bepaal $$$\int 2 \sin{\left(2 t \right)}\, dt$$$.

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

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

Zij $$$u=2 t$$$.

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

De integraal kan worden herschreven als

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

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 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

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

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

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

$$- \cos{\left({\color{red}{u}} \right)} = - \cos{\left({\color{red}{\left(2 t\right)}} \right)}$$

Dus,

$$\int{2 \sin{\left(2 t \right)} d t} = - \cos{\left(2 t \right)}$$

Voeg de integratieconstante toe:

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

$$$\int 2 \sin{\left(2 t \right)}\, dt = - \cos{\left(2 t \right)} + C$$$A