TITLE: Proof of Fresnel's Integral
QUESTION [8 upvotes]: So, my teacher wants us to prove the Fresnel's integral:
$$\int_0^\infty\cos(x^2)dx=\sqrt{\frac{\pi}{8}}$$
The problem is that we cannot use complex analysis to prove that and we should do that using the Euler identity:
$$\int_0^\infty\cos(x^2)dx=\frac{1}{2}\int_0^\infty e^{iw^2}dw+\frac{1}{2}\int_0^\infty e^{-iw^2}dw$$ 
But I have that integral of $e^{-iw^2}$ and I cannot solve that :(
I am an engineering student, so basically, I only have the "basic" calculus, just simple/double/triple integrals, some notions about series, ODE's and PDE's, but nothing as deep as in the regular Math degree, so probably there's no need to use hard stuff to figure this out.

REPLY [3 votes]: After using Euler's, use Gaussian Integral:
$$\int_{-\infty}^{\infty}e^{-a(x+b)^2}dx=\sqrt{
\frac{\pi}{a}}$$
or Integral of a Gaussian Function:
$$\int_{-\infty}^{\infty}ae^{-(x-b)^2/2c^2}dx=\sqrt{2}a\,|c|\sqrt{\pi}$$