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

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

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

Then $$$du=\left(\sqrt{5} x\right)^{\prime }dx = \sqrt{5} dx$$$ (steps can be seen »), and we have that $$$dx = \frac{\sqrt{5} du}{5}$$$.

So,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{\sqrt{5}}{5}$$$ and $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

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

This integral (Fresnel Cosine Integral) does not have a closed form:

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

Recall that $$$u=\sqrt{5} x$$$:

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

Therefore,

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

Add the constant of integration:

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

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