Integral de $$$\cos{\left(\omega t^{2} \right)}$$$ em relação a $$$t$$$

Encontre $$$\int \cos{\left(\omega t^{2} \right)}\, dt$$$.

Seja $$$u=\sqrt{\omega} t$$$.

Então $$$du=\left(\sqrt{\omega} t\right)^{\prime }dt = \sqrt{\omega} dt$$$ (veja os passos »), e obtemos $$$dt = \frac{du}{\sqrt{\omega}}$$$.

A integral torna-se

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{\sqrt{\omega}}$$$ e $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

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

Esta integral (Integral de Fresnel do cosseno) não possui forma fechada:

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

Recorde que $$$u=\sqrt{\omega} t$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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