calculus - How to prove that $int_0^{pi} log(|sin t|),textrm{dt} ;;textrm{is integrable }$

How to prove that

$$\int_0^{\pi} \log(|\sin t|)\,\textrm{dt} \;\;\textrm{is integrable }$$

Any hints would be appreciated.

Answer

Hint: It is true that

$$\sin{t} \ge \frac{2}{\pi} t$$ on the interval $[0, \pi/2]$; try drawing the graph, or noting that $\sin{t}$ is concave down and we have equality at the endpoints. Now show that

$$\int_0^{\pi/2} |\ln{\frac{2}{\pi} t}| dt < \infty$$

Do something similar for the right-half of the interval.