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

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

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

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

Deze integraal (Fresnel-cosinusintegraal) heeft geen gesloten vorm:

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

Dus,

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

Voeg de integratieconstante toe:

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

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