Integral de $$$108 \cos{\left(x^{2} \right)}$$$

Encontre $$$\int 108 \cos{\left(x^{2} \right)}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=108$$$ e $$$f{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:

$${\color{red}{\int{108 \cos{\left(x^{2} \right)} d x}}} = {\color{red}{\left(108 \int{\cos{\left(x^{2} \right)} d x}\right)}}$$

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

$$108 {\color{red}{\int{\cos{\left(x^{2} \right)} d x}}} = 108 {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)}{2}\right)}}$$

Portanto,

$$\int{108 \cos{\left(x^{2} \right)} d x} = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)$$

Adicione a constante de integração:

$$\int{108 \cos{\left(x^{2} \right)} d x} = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right)+C$$

$$$\int 108 \cos{\left(x^{2} \right)}\, dx = 54 \sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} x}{\sqrt{\pi}}\right) + C$$$A