Updated: Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$ using Fresnel Integrals Prove completely $\int^\infty_0 \cos(x^2)dx=\frac{\sqrt{2\pi}}{4}$
I've tried substituting $ x^2 = t $ but could not proceed at all thereafter in integration. Any help would be appreciated.
I should mentioned at the start that I am trying to use Fresnel Integrals.
That's why I was trying to substitute t=x^2 since I'm nearly positive that is the first step. However, thereafter I am lost.

Define $I(a)=\int_0^{\infty}\cos ax^2 dx$ for $a>0$. Note that $I(-a)=I(a)$. Now take a Laplace transform with respect to $a$ to obtain
\begin{align}
\mathcal{L}_{a \rightarrow s}\{I(a)\}&=\int_0^{\infty}\frac{s}{s^2+x^4} dx\\
&=\frac{\pi}{2\sqrt{2s}}.
\end{align}
Now take inverse Laplace transform to obtain 
\begin{align}
I(a)&=\mathcal{L}^{-1}_{s \rightarrow a}\{\frac{\pi}{2\sqrt{2s}}\}\\
&=\frac14\sqrt{\frac{2\pi}{a}}.
\end{align}
For your problem we then have $I(1)=\frac{\sqrt{2\pi}}{4}$.
