TITLE: Show that the integral $\int_{0}^{\pi/2} {\log \sin x} \,dx$ is convergent
QUESTION [5 upvotes]: Show that this is a convergent integral: $$\int_{0}^{\pi/2} {\log \sin x} \,dx$$

My approach: I was applying the limit test (refer below), but I am getting the value of $k$ as $0$. 

Limit Comparison Theorem (Type II): If $f$ and $g$ are continuous, positive functions for all values of $x$, and
  $$\lim\limits_{x\to a}\frac{f(x)}{g(x)}=k$$
  if $k$ is nonzero finite, then both integrals 
  $$\int_{a}^{b} {f(x)}\,dx$$ and $$\int_{a}^{b} {g(x)}\,dx$$ either converge or do not converge.

REPLY [6 votes]: Hint
Near $0^+$, the integrand has a constant sign.
$$\ln (\sin (x))\approx \ln (x) $$
$$\int_X^1\ln (t)\,dt=-1-X\ln (X)+X $$
and
$$\lim_{X\to 0^+}X\ln (X )=0$$