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