Answer
$$\frac{1}{2}\ln \left| y \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{1}{{2y}}} dy \cr
& {\text{take out the constant}} \cr
& = \frac{1}{2}\int {\frac{1}{y}} dy \cr
& {\text{integrate}} \cr
& = \frac{1}{2}\left( {\ln \left| y \right|} \right) + C \cr
& = \frac{1}{2}\ln \left| y \right| + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dy}}\left( {\frac{1}{2}\ln \left| y \right| + C} \right) \cr
& {\text{ = }}\frac{d}{{dy}}\left( {\frac{1}{2}\ln \left| y \right|} \right) + \frac{d}{{dx}}\left( C \right) \cr
& {\text{ = }}\frac{1}{2}\left( {\frac{1}{y}} \right) + 0 \cr
& = \frac{1}{{2y}} \cr} $$
