I want to compute the Fresnel integrals $$\int_{-\infty}^{\infty} e^{\pm i\cdot x^2}, \hspace{5mm} dx \int_{-\infty}^{\infty} \cos(x^2)dx, \hspace{5mm} \int_{-\infty}^{\infty} \sin(x^2)dx$$ using real analysis techniques, specifically parametric Integrals and differential Equations.
As a first step, I want to justify pulling a differentiation with respect to the parameter $s$ into a specific integral, like so: $$\frac{d}{ds} \int_{0}^{\infty} e^{-e^{is}\cdot x^2} dx = \int_{0}^{\infty} \frac{d}{ds} e^{-e^{is}\cdot x^2} dx.$$ I belive I have to show uniform integrability and continuous differentiability, but the I don't know how to get the technical details right.