In attempt to evaluate $$\text P\int^\infty_{-\infty}\frac1{x^2}dx$$ we consider $$\oint_C\frac{1}{z^2}dz$$ where $C$ is an infinitely large semicircle on the upper half plane centered at the origin, with a small indent at the origin.
Obviously, the indent integral and the large arc integral tends to zero. By Cauchy’s integral theorem, the contour integral is also zero. Thus, $$\text P\int^\infty_{-\infty}\frac1{x^2}dx=0$$
But the integrand is always positive. How could the integral equal zero? How can this counter-intuitive result be explained?