The integral {I=\int_0^\pi \log \sin x dx} is normally computed (e.g. in Ahlfors’ book) to be {-\pi \log 2} using complex integration over a suitable almost-rectangular contour. There is also a simple and direct way to get the value of this integral by a substitution and elementary calculus.

First, by the substitution {x=2t} and the identity {\sin(2x)=2\sin x \cos x},

\displaystyle  I = 2 \int_0^{\pi/2} \log \sin t dt + 2 \int_0^{\pi/2} \log \cos t dt + \pi \log 2;

then using the symmetry of {\sin} and {\cos} gives:

\displaystyle  I = I + I + \pi \log 2,

whence the result. There are slight technicalities regarding the improperness of these integrals, but they can be directly justified (or one may use the Lebesgue integral).

[Edit (7/25)- Todd Trimble posted solutions to similar integrals, which use the result of this post as a lemma, here.  AM]

So, in a break from the earlier series I was doing on Lie algebras, I want to discuss a very elementary question about polynomials. The answer is well-known but is interesting.   It would make a good competition type problem (indeed, it’s an exercise in Serge Lang’s Algebra).  Moreover, ironically, it’s useful in algebra: At some point one of us will probably discuss Hilbert polynomials, which take integer values, so this result tells us something about them.

We have a polynomial {P(X) \in \mathbb{Q}[X]} which takes integer values at all sufficiently large {n \in \mathbb{N}}. What can we say about {P}?

Denote the set of such {P} by {\mathfrak{I}}. Then clearly {\mathbb{Z}[X] \subset \mathfrak{I}}. But the converse is false.