Answer
$$\frac{1}{{2t}} - \sqrt 2 {e^t}$$
Work Step by Step
$$\eqalign{
& \int {\left[ {\frac{1}{{2t}} - \sqrt 2 {e^t}} \right]} dt \cr
& {\text{Split the integrand}} \cr
& = \int {\frac{1}{{2t}}} d - \int {\sqrt 2 {e^t}} dt \cr
& {\text{drop out constants}} \cr
& = \frac{1}{2}\int {\frac{1}{t}} dt - \sqrt 2 \int {{e^t}} dt \cr
& {\text{Integrate using basic rules}} \cr
& = \frac{1}{2}\ln \left| t \right| - \sqrt 2 {e^t} + C \cr
& \cr
& {\text{Checking by differentiation}} \cr
& \frac{d}{{dt}}\left[ {\frac{1}{2}\ln \left| t \right| - \sqrt 2 {e^t} + C} \right] \cr
& \frac{d}{{dt}}\left[ {\frac{1}{2}\ln \left| t \right|} \right] - \frac{d}{{dt}}\left[ {\sqrt 2 {e^t}} \right] + \frac{d}{{dt}}\left[ C \right] \cr
& \frac{1}{2}\frac{d}{{dt}}\left[ {\ln \left| t \right|} \right] - \sqrt 2 \frac{d}{{dt}}\left[ {{e^t}} \right] + \frac{d}{{dt}}\left[ C \right] \cr
& \frac{1}{2}\left( {\frac{1}{t}} \right) - \sqrt 2 \left( {{e^t}} \right) + 0 \cr
& \frac{1}{{2t}} - \sqrt 2 {e^t} \cr} $$
