Non-Elementary Integration · Hard problem

Let $I(a)=\displaystyle\int_0^{\infty}e^{-x^2}\cos(ax)\,dx$, so $I(0)=\dfrac{\sqrt\pi}{2}$. By differentiating under the integral sign, find $I(2)=\displaystyle\int_0^{\infty}e^{-x^2}\cos 2x\,dx$.