Integral de $$$\cos{\left(\frac{x^{2}}{18} \right)}$$$

Encontre $$$\int \cos{\left(\frac{x^{2}}{18} \right)}\, dx$$$.

Seja $$$u=\frac{\sqrt{2} x}{6}$$$.

Então $$$du=\left(\frac{\sqrt{2} x}{6}\right)^{\prime }dx = \frac{\sqrt{2}}{6} dx$$$ (veja os passos »), e obtemos $$$dx = 3 \sqrt{2} du$$$.

Assim,

$${\color{red}{\int{\cos{\left(\frac{x^{2}}{18} \right)} d x}}} = {\color{red}{\int{3 \sqrt{2} \cos{\left(u^{2} \right)} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=3 \sqrt{2}$$$ e $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

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

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

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

Recorde que $$$u=\frac{\sqrt{2} x}{6}$$$:

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

Portanto,

$$\int{\cos{\left(\frac{x^{2}}{18} \right)} d x} = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right)$$

Adicione a constante de integração:

$$\int{\cos{\left(\frac{x^{2}}{18} \right)} d x} = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right)+C$$

$$$\int \cos{\left(\frac{x^{2}}{18} \right)}\, dx = 3 \sqrt{\pi} C\left(\frac{x}{3 \sqrt{\pi}}\right) + C$$$A