Some integrals look simple enough to try with familiar calculus techniques, then resist every straightforward approach. This project began with one such problem, proposed by Brian Bradie in the College Mathematics Journal: integrate arctan(x)/(1 + αx + x²) from zero to infinity, for α > -2.
I led a three-person team with Evan Pfeifer and Adrian Hito through NCSSM’s Research in Mathematics program. We wanted a formula that worked across the parameter α, rather than a separate numerical answer for each value.
Our approach and what we learned
We started with numerical exploration and special cases, then moved into complex analysis. By rewriting arctangent using a complex logarithm, we could examine the integral along paths in the complex plane. We compared contour geometries and worked with the dilogarithm, a special function that appeared in the resulting expressions.
The May 2024 report documents useful reformulations and the limits of the approaches we tried. We did not finish a simplified general evaluation of the definite integral. The value of the project was learning how to move between numerical evidence, symbolic reasoning, and proofs, including recognizing where a promising method still needed more work.