Answer
$$t + \ln \left| t \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{t + 1}}{t}dt} \cr
& = \int {\left( {\frac{t}{t} + \frac{1}{t}} \right)dt} \cr
& = \int {\left( {1 + \frac{1}{t}} \right)dt} \cr
& {\text{sum rule}} \cr
& = \int {dt} + \int {\frac{1}{t}dt} \cr
& {\text{integrate}} \cr
& = t + \ln \left| t \right| + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dt}}\left( {t + \ln \left| t \right| + C} \right) \cr
& {\text{ = }}\frac{d}{{dt}}\left( t \right) + \frac{d}{{dt}}\left( {\ln \left| t \right|} \right) + \frac{d}{{dt}}\left( C \right) \cr
& {\text{ = }}1 + \frac{1}{t} + 0 \cr
& add \cr
& = \frac{{t + 1}}{t} \cr} $$
