a)\[\int {\frac{{{{\cos }^2}x}}{{1 - \sin {\rm{x}}}}dx}  = \int {\frac{{1 - {{\sin }^2}x}}{{1 - \sin {\rm{x}}}}dx} \]

                        \[ = \int {\frac{{\left( {1 - \sin {\rm{x}}} \right)\left( {1 + \sin {\rm{x}}} \right)}}{{\left( {1 - \sin {\rm{x}}} \right)}}dx} \]

                        \[ = \int {\left( {1 + \sin {\rm{x}}} \right)dx}  = x - \cos x + C.\]

b) \[\int {\left( {1 + 3{{\sin }^2}\frac{x}{2}} \right)dx}  = \int {\left( {1 + 3.\frac{{1 - \cos x}}{2}} \right)dx} \]

                                \[ = \int {\left( {\frac{5}{2} - \frac{3}{2}\cos x} \right)dx} \]

                                \[ = \frac{5}{2}x - \frac{3}{2}\sin {\rm{x}} + C\].

c) \[\int {\frac{{2{{\cos }^3}x + 3}}{{{{\cos }^2}x}}dx}  = \int {\left( {2\cos x + \frac{3}{{{{\cos }^2}x}}} \right)dx} \]

                              \[ = \int {2\cos xdx + \int {\frac{3}{{{{\cos }^2}x}}dx} } \]

                                               \[ = 2\sin x + 3\tan x + C\]