# On a special kind of integral

## TL;DR

This paper derives a closed-form expression for a specific improper integral involving logarithmic and rational functions, providing a formula useful for analytical and numerical evaluations in mathematical analysis.

## Contribution

The paper presents the first explicit closed-form solution for the integral of rac{ ext{ln}(x)}{x^n+1} over [0, ∞) for n > 1, linking it to special functions and derivatives of gamma functions.

## Key findings

- Derived a closed-form formula involving cotangent and cosecant functions.
- Expressed the integral as a derivative of gamma functions.
- Facilitated easier computation and approximation of similar integrals.

## Abstract

In the world of mathematical analysis, many counterintuitive answers arise from the manipulation of seemingly unrelated concepts, ideas, or functions. For example, Euler showed that $e^{i\pi} + 1 = 0$, whereas Gauss proved that the area underneath $y = e^{-x^2}$ spanning the whole real axis is $ \sqrt{\pi} $. In this paper, we will determine the closed-form solution of the improper integral \[ I_n = \int_{0}^{\infty} \frac{\ln{x}}{x^n+1} dx, \ \forall n \in \mathbb{R} \text{, with}\ n > 1. \] Determining closed-form solutions of improper integrals have real implications not only in easing the solving of similar, yet more difficult integrals, but also in speeding up numerical approximations of the answer by making them more efficient. Following our calculations, we derived the formula \[ I_n = \int_{0}^{\infty} \frac{\ln{x}}{x^n+1} dx = -\frac{\pi^2}{n^2}\cot{\frac{\pi}{n}}\csc{\frac{\pi}{n}} = -\frac{d}{dn} \Bigg[ \Gamma\Big(1-\frac{1}{n}\Big) \Gamma\Big(\frac{1}{n}\Big) \Bigg]. \]