Generalized Log-sine integrals and Bell polynomials

In this paper, we investigate the integral of xnlogp(sin(x)) for natural numbers n and p. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Lsn+p+1(n)(θ). Later, we use properties of Bell polynomials to find an expression for the derivative of the central binomial and shifted central binomial coefficients as finite sums of polygamma functions and harmonic numbers.