Answer
$\displaystyle \frac{2\sqrt{3}}{3}$
Work Step by Step
Use the table "Basic Integration Rules", p.246
$\displaystyle \frac{d}{dx}[\tan x]=\sec^{2}x$
$F(x)=\tan x,\qquad F^{\prime}(x)=f(x)$
(use FTC, Th.4.9 )
$\displaystyle \int_{-\pi/6}^{\pi/6}\sec^{2}xdx=F(\frac{\pi}{6})-F(-\frac{\pi}{6})$
$= [\tan x]_{-\pi/6}^{\pi/6}$
$=\displaystyle \frac{\sqrt{3}}{3}-(-\frac{\sqrt{3}}{3})$
$=\displaystyle \frac{2\sqrt{3}}{3}$
Check: using desmos.com (see image below)
