Evaluate $\displaystyle \int_0^\infty\cos x^2\,dx$
would this help you?? Fresnel Integrals -- from Wolfram MathWorld
Substitute $\displaystyle u=x^2,$ the integral becomes to $\displaystyle \int_0^\infty {\frac{{\cos u}}{{2\sqrt u }}\,du} .$
The following parameter may be useful: $\displaystyle \int_0^\infty {\frac{{e^{ - \alpha u} }}{{\sqrt \alpha }}\,d\alpha } = \frac{{\sqrt \pi }}{{\sqrt u }}.$
To prove this, we can use a substitution and the remarkable fact $\displaystyle \int_0^\infty {e^{ - x^2 } \,dx} = \frac{{\sqrt \pi }}{2}.$
Now plug the parameter into the integral, so we'll have a double integral which can be computed without problems.