Integral de $$$\cos{\left(5 x^{2} \right)}$$$

Halla $$$\int \cos{\left(5 x^{2} \right)}\, dx$$$.

Sea $$$u=\sqrt{5} x$$$.

Entonces $$$du=\left(\sqrt{5} x\right)^{\prime }dx = \sqrt{5} dx$$$ (los pasos pueden verse »), y obtenemos que $$$dx = \frac{\sqrt{5} du}{5}$$$.

Por lo tanto,

$${\color{red}{\int{\cos{\left(5 x^{2} \right)} d x}}} = {\color{red}{\int{\frac{\sqrt{5} \cos{\left(u^{2} \right)}}{5} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{\sqrt{5}}{5}$$$ y $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

$${\color{red}{\int{\frac{\sqrt{5} \cos{\left(u^{2} \right)}}{5} d u}}} = {\color{red}{\left(\frac{\sqrt{5} \int{\cos{\left(u^{2} \right)} d u}}{5}\right)}}$$

Esta integral (Integral del coseno de Fresnel) no tiene una forma cerrada:

$$\frac{\sqrt{5} {\color{red}{\int{\cos{\left(u^{2} \right)} d u}}}}{5} = \frac{\sqrt{5} {\color{red}{\left(\frac{\sqrt{2} \sqrt{\pi} C\left(\frac{\sqrt{2} u}{\sqrt{\pi}}\right)}{2}\right)}}}{5}$$

Recordemos que $$$u=\sqrt{5} x$$$:

$$\frac{\sqrt{10} \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{u}}}{\sqrt{\pi}}\right)}{10} = \frac{\sqrt{10} \sqrt{\pi} C\left(\frac{\sqrt{2} {\color{red}{\sqrt{5} x}}}{\sqrt{\pi}}\right)}{10}$$

Por lo tanto,

$$\int{\cos{\left(5 x^{2} \right)} d x} = \frac{\sqrt{10} \sqrt{\pi} C\left(\frac{\sqrt{10} x}{\sqrt{\pi}}\right)}{10}$$

Añade la constante de integración:

$$\int{\cos{\left(5 x^{2} \right)} d x} = \frac{\sqrt{10} \sqrt{\pi} C\left(\frac{\sqrt{10} x}{\sqrt{\pi}}\right)}{10}+C$$

$$$\int \cos{\left(5 x^{2} \right)}\, dx = \frac{\sqrt{10} \sqrt{\pi} C\left(\frac{\sqrt{10} x}{\sqrt{\pi}}\right)}{10} + C$$$A