# Generalized Log-sine integrals and Bell polynomials

## TL;DR

This paper explores integrals involving powers and logarithms of sine, connects them to log-sine integrals, and employs Bell polynomials to derive closed-form expressions for derivatives of binomial coefficients.

## Contribution

It introduces a unified approach to log-sine integrals and applies Bell polynomials to obtain new closed-form formulas for binomial coefficient derivatives.

## Key findings

- Derived expressions for integrals of $x^n ext{log}^m( ext{sin}(x))$
- Connected log-sine integrals to known special functions
- Provided closed-form formulas for derivatives of binomial coefficients

## Abstract

In this paper, we investigate the integral of $x^n\log^m(\sin(x))$ for natural numbers $m$ and $n$. In doing so, we recover some well-known results and remark on some relations to the log-sine integral $\operatorname{Ls}_{n+m+1}^{(n)}(\theta)$. Later, we use properties of Bell polynomials to find a closed expression for the derivative of the central binomial and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers.