  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.
