Answer
$$\theta - \cos \theta + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {1 + {{\sin }^2}\theta \csc \theta } \right)d\theta } \cr
& {\text{basic trigonometric identities }}\csc \theta = \frac{1}{{\sin \theta }} \cr
& = \int {\left( {1 + {{\sin }^2}\theta \left( {\frac{1}{{\sin \theta }}} \right)} \right)d\theta } \cr
& = \int {\left( {1 + \sin \theta } \right)d\theta } \cr
& = \int {d\theta } + \int {\sin } d\theta \cr
& {\text{use power rule and integration formulas from table 4}}{\text{.2}}{\text{.1}} \cr
& = \theta + \left( { - \cos \theta } \right) + C \cr
& = \theta - \cos \theta + C \cr} $$
