Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 3 / b

The Gaussian function $e^{-x^2}$ is even and $\sin x$ is odd, so their product is an odd function. Moreover $|e^{-x^2}\sin x|\le e^{-x^2}$, whose integral on the real line is finite. Thus the improper integral is absolutely convergent, not merely a symmetric principal value, and the contributions of its two half-lines cancel: