Estimation the integration of $\frac{x}{\sin (x)} $ How to prove that integral of $\frac{x}{\sin (x)} $ between $0$ to $\frac{\pi}{2}$ lies within the interval $\frac{\pi^2}{4}$ and $\frac{\pi}{2}$ ?

Recall from elementary geometry that the sine function satisfies the inequalities
$$ \frac{2}{\pi}x \le \sin(x)\le x \tag 1$$
for $0\le x\le \pi/2$.  
We see immediately from $(1)$ that 
$$1 \le \frac{x}{\sin(x)}\le \frac{\pi}{2} \tag 2$$
for $0<x\le \pi/2$ from which the inequality of interest is trivial.  
