Answer
$$F\left( t \right) = 2\ln \left| u \right| + \frac{{{u^2}}}{2} + \frac{9}{2}$$
Work Step by Step
$$\eqalign{
& f\left( u \right) = \frac{2}{u} + u;\,\,\,F\left( 1 \right) = 5 \cr
& {\text{Write a formula }}F\left( u \right){\text{ for the antiderivative of }}f\left( u \right) \cr
& F\left( u \right) = \int {\left( {\frac{2}{u} + u} \right)} du \cr
& {\text{integrate}} \cr
& F\left( t \right) = 2\ln \left| u \right| + \frac{{{u^2}}}{2} + C \cr
& \cr
& {\text{Use the condition }}F\left( 1 \right) = 5{\text{ to find }}C \cr
& 5 = 2\ln \left| 1 \right| + \frac{{{1^2}}}{2} + C \cr
& 5 = 0 + \frac{1}{2} + C \cr
& C = \frac{9}{2} \cr
& \cr
& {\text{The specific antiderivative of }}f{\text{ is}} \cr
& F\left( t \right) = 2\ln \left| u \right| + \frac{{{u^2}}}{2} + \frac{9}{2} \cr} $$
