Integrate $\int_{-\infty}^{\infty} e^{itx}\frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx?$

Question

This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He mentioned that this sort of integral can be generally solved through complex analysis but did not elaborate. Can someone show me how?