Answer
$\frac{\pi}{12}\leq \int_{\pi/4}^{\pi/3}\tan x dx\leq \frac{\sqrt 3\pi}{12}$
Work Step by Step
$$\frac{\pi}{4} \leq x \leq \frac{\pi}{3}$$
The tangent function is increasing on the given domain:
$$\tan\frac{\pi}{4}\leq \tan x \leq \tan\frac{\pi}{3}$$
$$1 \leq \tan x \leq \sqrt 3$$
Using the property $8$ it follows:
$$1\left(\frac{\pi}{3}-\frac{\pi}{4}\right) \leq \int_{\pi/4}^{\pi/3}\tan x dx\leq \sqrt 3\left(\frac{\pi}{3}-\frac{\pi}{4}\right)$$
$$\frac{\pi}{12}\leq \int_{\pi/4}^{\pi/3}\tan x dx\leq \frac{\sqrt 3\pi}{12}$$
