integration - Fresnel Integral?

Is $\int\cos\left(\frac{\pi}{2}\cdot x^2\right)\,dx$ a known integral ?

I found on the net something called Fresnel integral, but we didn't learn it, and it also somehow related to Euler, and we didn't touch the Euler stuff, so maybe I made a mistake before it while calculating the double integral:

$$\iint_D\sin\left(\frac{\pi x}{2y}\right)\,dxdy$$ where $D=\left\{(x,y)\in\mathbb R^2: x\le y\le \sqrt[3]{x}\wedge y\ge\frac{\sqrt{2}}{2}\right\}$

So I wrote the $D$ as a simple to $x$:

$\begin{cases}\frac{\sqrt{2}}{2}\le y\le 1\\y^3\le x\le y\end{cases}$

and then I did the integral by $x$ and got that troublesome integral.

So can I solve this problem without Fresnel integral or maybe i have some mistake on the way?

Answer

Yes, you can do without Fresnel integrals I give you the first step:
$$\int_{y^3}^y \sin\Bigl(\frac{\pi x}{2y}\Bigr)\,dx = \frac{2}{\pi}y\cos\Bigl(\frac{\pi y^2}{2}\Bigr).$$
Then integrate with respect to $y$ (if you do not see how to do that I give you the hint to change variable $u=y^2$).