I am trying to find an integral:
$$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$
I went about applying contour integral over a semicircle with diameter along $ x = +\infty$ to $- \infty $ enclosing the pole at $x = +i $. The residue is $(-i/2)$ as shown here.
So the integral should be $(2\pi i)\times (-i/2)=\pi$
However since it is a well behaved function if I do a quick numerical integration in Mathematica it is giving me a value of $0.475$
A plot confirms that the function converges very fast.
Mathematica code:
NIntegrate[ E^{- Sqrt[1 + y^2]}/(1 + y^2)^2, {y, -1000, 1000}]
What am I doing wrong?
Many thanks!