# How I Impressed My Wife: Part 1

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

$\displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

Yes, I married well indeed.

This integral serves as the theoretical underpinnings for finding vortices in the velocity fields of atomic wave functions and can be obtained from Equation (4) of a 2014 paper in Physical Review A that she co-authored. And I really like this integral because there are so many different ways of evaluating it, including various trigonometric identities, the magic substitution $u = \tan (x/2)$, partial fractions, and even contour integration and residues.

In this series, I’ll explore different ways of evaluating this integral, starting tomorrow. Until then, I offer a thought bubble if you’d like to try to tackle this before I present the solution(s).