Is the integral $$I(n) = \int_0^{\pi} \sin^{n}(x) \ln(\sin(x)) dx$$ analytically tractable for $n \in \mathbb{N}$? If not, are there good upper and lower bounds?
To clarify, although this has a fairly standard-looking form, I haven't been able to find an answer anywhere and I've tried a few of the usual tricks (although I'm a tad rusty).
I have tried integration by parts to reduce to a simpler or recursive form but this seems to make the expression more complex, and likewise with the substitution $u=\sin(x)$.
Equally, it is relatively easy to see that the expression under the integral is non-positive. Taking a derivative $$ \frac{d}{dx} \sin^{n}(x) \ln(\sin(x)) = \sin^{n-1}(x) \cos(x) (1 + n \ln(\sin(x))) = 0$$ gives a has minimum at $-\frac{1}{en}$.
Together these give the reasonably simple bounds: $$-\frac{\pi}{en} < I(n) < 0.$$
But numerically these don't appear to be particularly tight, so I'm still hopeful of an exact solution or an improvement.