Answer
$$\frac{1}{2}\ln \left| t \right| + {t^2} + C$$
Work Step by Step
$$\eqalign{
& {\text{Evaluate }}\int {\left[ {\frac{1}{{2t}} + 2t} \right]} dt \cr
& {\text{Sum rule for integration}} \cr
& = \int {\frac{1}{{2t}}} dt + \int {2t} dt \cr
& = \frac{1}{2}\int {\frac{1}{t}} dt + 2\int t dt \cr
& {\text{Integration basic rules}} \cr
& = \frac{1}{2}\ln \left| t \right| + 2\left( {\frac{{{t^2}}}{2}} \right) + C \cr
& {\text{simplify}} \cr
& = \frac{1}{2}\ln \left| t \right| + {t^2} + C \cr
& \cr
& {\text{Checking by differentiation}} \cr
& \frac{d}{{dx}}\left[ {\frac{1}{2}\ln \left| t \right| + {t^2} + C} \right] \cr
& = \left( {\frac{1}{{2t}}} \right) + 2t + 0 \cr
& = \frac{1}{{2t}} + 2t \cr} $$
