Integraal van $$$\cos{\left(3 x^{2} \right)}$$$

Bepaal $$$\int \cos{\left(3 x^{2} \right)}\, dx$$$.

Zij $$$u=\sqrt{3} x$$$.

Dan $$$du=\left(\sqrt{3} x\right)^{\prime }dx = \sqrt{3} dx$$$ (de stappen zijn te zien »), en dan geldt dat $$$dx = \frac{\sqrt{3} du}{3}$$$.

De integraal kan worden herschreven als

$${\color{red}{\int{\cos{\left(3 x^{2} \right)} d x}}} = {\color{red}{\int{\frac{\sqrt{3} \cos{\left(u^{2} \right)}}{3} d u}}}$$

Pas de constante-veelvoudregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ toe met $$$c=\frac{\sqrt{3}}{3}$$$ en $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

$${\color{red}{\int{\frac{\sqrt{3} \cos{\left(u^{2} \right)}}{3} d u}}} = {\color{red}{\left(\frac{\sqrt{3} \int{\cos{\left(u^{2} \right)} d u}}{3}\right)}}$$

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

$$\frac{\sqrt{3} {\color{red}{\int{\cos{\left(u^{2} \right)} d u}}}}{3} = \frac{\sqrt{3} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} u}{\sqrt{\pi}}\right)}{2}\right)}}}{3}$$

We herinneren eraan dat $$$u=\sqrt{3} x$$$:

$$\frac{\sqrt{6} \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{u}}}{\sqrt{\pi}}\right)}{6} = \frac{\sqrt{6} \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{\sqrt{3} x}}}{\sqrt{\pi}}\right)}{6}$$

Dus,

$$\int{\cos{\left(3 x^{2} \right)} d x} = \frac{\sqrt{6} \sqrt{\pi} C\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6}$$

Voeg de integratieconstante toe:

$$\int{\cos{\left(3 x^{2} \right)} d x} = \frac{\sqrt{6} \sqrt{\pi} C\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6}+C$$

$$$\int \cos{\left(3 x^{2} \right)}\, dx = \frac{\sqrt{6} \sqrt{\pi} C\left(\frac{\sqrt{6} x}{\sqrt{\pi}}\right)}{6} + C$$$A