I was differentiating a completely different integral when I came across $$I(\alpha,\beta,n)=\int_0^\frac{\pi}{2} \sin^\alpha(nt) \cos^\beta(t)dt.$$
Evidently, $I(\alpha, \beta,1)= \frac{1}{2} B \left(\frac{\alpha+1}{2},\frac{\beta+1}{2} \right)$, where $B$ is the Beta function. Is this function expressible in terms of other widely-known functions? I doubt Chebyschev Polynomials will help here.