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.

As is common, use $f(z)=e^{-iz^2}=\cos(z^2)-i\sin(z^2)$
Now $$\int_{-\infty}^{\infty}e^{-iz^2}{\rm d}z=\int_{-\infty}^{\infty}e^{-\left(e^{i\pi/4}z\right)^2}{\rm d}z=\frac1{e^{i\pi/4}}\int_{-\infty}^{\infty}e^{-x^2}{\rm d}x=e^{-i\pi/4}\sqrt{\pi}=\sqrt{\frac{\pi}2}-i\sqrt{\frac{\pi}2}$$
Now, since $f(z)$ is even:
$$\int_0^{\infty}\cos(x^2){\rm d}x=\Re\left(\frac12\int_{-\infty}^{\infty}e^{-iz^2}{\rm d}z\right)=\frac12\sqrt{\frac{\pi}2}=\frac{\sqrt{2\pi}}4$$ 
