Answer
$$\int e^x\sqrt{1+e^x}dx=\frac{2}{3}(1+e^x)^{3/2}+c$$
Work Step by Step
To evaluate the integral $$\int e^x\sqrt{1+e^x}dx$$
we will use substitution $e^x=t$ which gives us $e^xdx=dt$. Putting this into the integral we get:
$$\int \sqrt{1+e^x}e^xdx=\int\sqrt tdt=\int t^{1/2}dt=\frac{t^{3/2}}{\frac{3}{2}}+c=\frac{2}{3}t^{3/2}+c$$
where $c$ is arbitrary constant. Now we have to express solution in terms of $x$:
$$\int e^x\sqrt{1+e^x}dx=\frac{2}{3}t^{3/2}+c=\frac{2}{3}(1+e^x)^{3/2}+c$$
