Answer
$$1$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\frac{{d\theta }}{{1 + \sin 2\theta }}} \cr
& {\text{A matching integral in a table of integrals at the end of the book is the }} \cr
& {\text{formula }}44 \cr
& \int {\frac{{dx}}{{1 + \sin ax}}dx = - \frac{1}{a}\tan \left( {\frac{\pi }{4} - \frac{{ax}}{2}} \right) + C} \cr
& {\text{ }}a = 2 \cr
& then \cr
& \int_0^{\pi /2} {\frac{{d\theta }}{{1 + \sin 2\theta }}} = \left( { - \frac{1}{2}\tan \left( {\frac{\pi }{4} - \frac{{2x}}{2}} \right)} \right)_0^{\pi /2} \cr
& = - \frac{1}{2}\left( {\tan \left( {\frac{\pi }{4} - x} \right)} \right)_0^{\pi /2} \cr
& {\text{evaluate limits}} \cr
& = - \frac{1}{2}\left( {\tan \left( { - \frac{\pi }{4}} \right)} \right) + \frac{1}{2}\left( {\tan \left( {\frac{\pi }{4}} \right)} \right) \cr
& = - \frac{1}{2}\left( {\left( { - 1} \right)} \right) + \frac{1}{2}\left( {\left( 1 \right)} \right) \cr
& = 1 \cr} $$
