Answer
$$\frac{1}{6}{\left( {3 + 4{e^\theta }} \right)^{3/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {{e^\theta }\sqrt {3 + 4{e^\theta }} } d\theta \cr
& {\text{Integrate by using the substitution method}} \cr
& \,\,\,{\text{Let }}u = 3 + 4{e^\theta },\,\,\,\,du = 4{e^\theta }d\theta ,\,\,\,d\theta = \frac{{du}}{{4{e^\theta }}} \cr
& {\text{Write the integrand in terms of }}u \cr
& \int {{e^\theta }\sqrt {3 + 4{e^\theta }} } d\theta = \int {{e^\theta }\sqrt u } \left( {\frac{{du}}{{4{e^\theta }}}} \right) \cr
& = \frac{1}{4}\int {\sqrt u } du \cr
& = \frac{1}{4}\left( {\frac{{{u^{3/2}}}}{{3/2}}} \right) + C \cr
& = \frac{1}{6}{u^{3/2}} + C \cr
& \cr
& {\text{Write in terms of }}\theta ;{\text{ substitute }}3 + 4{e^\theta }{\text{ for }}u \cr
& = \frac{1}{6}{\left( {3 + 4{e^\theta }} \right)^{3/2}} + C \cr} $$
