How to decide convergence of Integrals

Notice, however, that

\begin{eqnarray} \lim_{x\to0^+}\ln(\sin x)\sin x&=&\lim_{x\to0^+}\frac{\ln(\sin x)}{\csc x}\\ &=&\lim_{x\to0^+}\frac{\cot x}{(-\csc x\cot x)}\\ &=&\lim_{x\to0^+}(-\sin x)\\ &=&0 \end{eqnarray}

Here is the graph of $y=\ln(\sin x)\sin x$