Proving $\int_0^1 \ln(\sin(1)/ \sin(x) )dx \leq 1$ using elementary calculus [closed]
Notice that \begin{align} x \sin(1) \le \sin x \end{align} for $x \in [0, 1]$. Then it follows that \begin{align} \int^1_0 \ln\left(\frac{\sin(1)}{\sin(x)} \right) dx \le \int^1_0 \ln\left(\frac{1}{x} \right)d x = 1. \end{align}