I want ask how to start to tackle the problems of inequalities involving integrals:

#1 Prove that

$\displaystyle 3\sqrt{e}\leq\int_{e}^{4e}\frac{lnx}{\sqrt{x}}dx \leq6$

#2 Prove that

$\displaystyle \frac{\pi}{2}\leq\int_{0}^{\frac{\pi}{2}} $$\displaystyle \frac{dx}{\sqrt{1-\frac{1}{2} \sin^2{x}}}$$\displaystyle \leq\frac{\pi}{\sqrt{2}}$

#3 Prove that

$\displaystyle 1\leq \int_{0}^{\frac{\pi}{2}} \frac{\sin{x}}{x} dx \leq \frac{\pi}{2}$