Answer
$$\frac{7}{6}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /6} {{{\sec }^3}2\theta } \tan 2\theta d\theta \cr
& {\text{Write the integrand as}} \cr
& = \int_0^{\pi /6} {{{\sec }^{2 + 1}}2\theta } \tan 2\theta d\theta \cr
& = \int_0^{\pi /6} {{{\sec }^2}2\theta } \sec 2\theta \tan 2\theta d\theta \cr
& = \frac{1}{2}\int_0^{\pi /6} {{{\sec }^2}2\theta } \left[ {\sec 2\theta \tan 2\theta \left( 2 \right)} \right]d\theta \cr
& {\text{Integrate}} \cr
& = \frac{1}{2}\left( {\frac{{{{\sec }^3}2\theta }}{3}} \right)_0^{\pi /6} \cr
& = \frac{1}{2}\left( {\frac{{{{\sec }^3}2\left( {\pi /6} \right)}}{3} - \frac{{{{\sec }^3}2\left( 0 \right)}}{3}} \right) \cr
& = \frac{1}{2}\left( {\frac{{{{\sec }^3}\left( {\pi /3} \right)}}{3} - \frac{{{{\sec }^3}\left( 0 \right)}}{3}} \right) \cr
& = \frac{1}{2}\left( {\frac{8}{3} - \frac{1}{3}} \right) \cr
& = \frac{1}{2}\left( {\frac{7}{3}} \right) \cr
& = \frac{7}{6} \cr} $$