Answer
$$6\cosh \left( {\frac{x}{2} + \ln 5} \right) + C $$
Work Step by Step
$$\eqalign{
& \int {3\sinh \left( {\frac{x}{2} + \ln 5} \right)} dx \cr
& {\text{integrate by the substitution method}} \cr
& {\text{set }}u = \frac{x}{2} + \ln 5{\text{ then }}\frac{{du}}{{dx}} = \frac{1}{2},\,\,\,\,dx = 2du \cr
& {\text{write the integrand in terms of }}u \cr
& \int {3\sinh \left( {\frac{x}{2} + \ln 5} \right)} dx = \int {3\sinh \left( u \right)} \left( {2du} \right) \cr
& {\text{cancel common terms}} \cr
& = 6\int {\sinh \left( u \right)} du \cr
& {\text{integrating}}{\text{,}} \cr
& = 6\cosh \left( u \right) + C \cr
& {\text{replace }}\frac{x}{2} + \ln 5{\text{ for }}u \cr
& = 6\cosh \left( {\frac{x}{2} + \ln 5} \right) + C \cr} $$
