Find

$$\int \frac{e^{2x}}{\sqrt{e^x+1}}dx$$

During a chemical reaction, the rate of change of the mass $m$ of a byproduct, in grams per hour, is modeled by the equation

$$\frac{dm}{dt}=\frac{27t}{\sqrt{3t+4}},t\ge 0$$

Use the substitution $u=\sqrt{3t+4}$ to show that

$$\int \frac{27t}{\sqrt{3t+4}}dt=2(3t+4{)}^{\frac{1}{2}}(At+B)+k$$

where $A$ and $B$ are integers to be found and $k$ is a constant of integration.