Expressing $\int dx\, \cos(ax^2 + 2bx + c)$ in terms of Fresnel integrals I have difficulties with this integral, which is is related to Fresnel integrals ($a>0$):
$$ \int dx\, \cos(ax^2 + 2bx + c) = \\ =\sqrt{\frac{\pi}{2a}} \left[ \cos(\frac{ac-b^2}{a}) \mathscr{C} \left( \frac{\sqrt{2} (ax + b)}{\sqrt{a\pi}} \right) -  \sin(\frac{ac-b^2}{a}) \mathscr{S}\left( \frac{\sqrt{2} (ax + b)}{\sqrt{a\pi}} \right) \right] +C $$ 
with $$ \mathscr{C}(x) = \int_0^x dt \cos(\pi t^2 /2), $$
$$ \mathscr{S}(x) = \int_0^x dt \sin(\pi t^2 /2). $$
How do you show this integral?

To demonstrate validity: Take derivatives and recall that 
$$\frac{d}{dx} \int_0^x f(t)dt=f(x)$$
and use that $$\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$$ 
For evaluation that is a different issue.
