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

Bepaal $$$\int \cos{\left(\frac{2 x}{\pi} \right)}\, dx$$$.

Zij $$$u=\frac{2 x}{\pi}$$$.

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

De integraal kan worden herschreven als

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

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

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

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

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

We herinneren eraan dat $$$u=\frac{2 x}{\pi}$$$:

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

Dus,

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

Voeg de integratieconstante toe:

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

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