Contour integral involving gamma function

[CAF123]

Homework Statement

Evaluate the integral by closing a contour in the complex plane $$\int_{-\infty}^{\infty} dx e^{iax^2/2}$$

Homework Equations

Residue theorem

The Attempt at a Solution

My initial choice of contour was a semicircle of radius R and a line segment from -R to R. In the limit R to infinity, I would hopefully recover the integral in OP. But then I realized this was going to work because the integral over the semicircle would not vanish when R tended to infinity. My question is, what is the best way to see what the best contour should be for a given problem and how to determine the one right for this problem?

Many thanks

 

[haruspex]

By symmetry, you could conside only a semi infinite integral.
Because of the exponential term, I suggest a rectangle will work better than an arc.