Answer
$$\ln 4$$
Work Step by Step
$$\eqalign{
& \int_{ - \pi /3}^{\pi /3} {\sqrt {{{\sec }^2}\theta - 1} d\theta } \cr
& \sqrt {{{\sec }^2}\theta - 1} {\text{ is an even function}} \cr
& {\text{use }}\int_{ - a}^a {f\left( x \right)dx = 2\int_0^a {f\left( x \right)dx,{\text{ }}f\left( x \right)} {\text{ is even}}} \cr
& \int_{ - \pi /3}^{\pi /3} {\sqrt {{{\sec }^2}\theta - 1} d\theta } = 2\int_0^{\pi /3} {\sqrt {{{\sec }^2}\theta - 1} d\theta } \cr
& {\text{identity }}{\sec ^2}\theta - 1 = {\tan ^2}\theta \cr
& = 2\int_0^{\pi /3} {\sqrt {{{\tan }^2}\theta } d\theta } \cr
& = 2\int_0^{\pi /3} {\tan \theta d\theta } \cr
& = 2\int_0^{\pi /3} {\frac{{\sin \theta }}{{\cos \theta }}d\theta } \cr
& {\text{integrating}} \cr
& = 2\left[ { - \ln \left| {\cos \theta } \right|} \right]_0^{\pi /3} \cr
& = - 2\left[ {\ln \left| {\cos \theta } \right|} \right]_0^{\pi /3} \cr
& {\text{evaluate limits}} \cr
& = - 2\left[ {\ln \left| {\cos \frac{\pi }{3}} \right| - \ln \left| {\cos 0} \right|} \right] \cr
& {\text{simplify}} \cr
& = - 2\left[ {\ln \left| {\frac{1}{2}} \right| - \ln \left| 1 \right|} \right] \cr
& = - 2\ln \left( {\frac{1}{2}} \right) \cr
& = \ln 4 \cr} $$
