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

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

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

Then $$$du=\left(\frac{\sqrt{2} x}{6}\right)^{\prime }dx = \frac{\sqrt{2}}{6} dx$$$ (steps can be seen »), and we have that $$$dx = 3 \sqrt{2} du$$$.

The integral becomes

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=3 \sqrt{2}$$$ and $$$f{\left(u \right)} = \cos{\left(u^{2} \right)}$$$:

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

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

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

Recall that $$$u=\frac{\sqrt{2} x}{6}$$$:

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

Therefore,

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

Add the constant of integration:

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

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