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

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

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

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

Zij $$$u=\pi t$$$.

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

De integraal kan worden herschreven als

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

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

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

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

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

We herinneren eraan dat $$$u=\pi t$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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