Answer
$$2\sin \left( {\sqrt t + 3} \right) + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{1}{{\sqrt t }}\cos \left( {\sqrt t + 3} \right)} dt \cr
& {\text{Integrate by the substitution method}} \cr
& u = \sqrt t + 3,\,\,\,\,\,du = \frac{1}{{2\sqrt t }}dt,\,\,\,dt = 2\sqrt t du \cr
& {\text{Write the integrand in terms of }}u \cr
& \int {\frac{1}{{\sqrt t }}\cos \left( {\sqrt t + 3} \right)} dt = \int {\frac{1}{{\sqrt t }}\cos \left( u \right)} \left( {2\sqrt t du} \right) \cr
& = \int {\cos \left( u \right)} \left( {2du} \right) \cr
& = 2\int {\cos u} du \cr
& = 2\sin u + C \cr
& {\text{Write in terms of }}t;{\text{ substitute }}\sqrt t + 3{\text{ for }}u \cr
& = 2\sin \left( {\sqrt t + 3} \right) + C \cr} $$
