Answer
$$\int e^x\sqrt{1+e^x}dx=\frac{2}{3}\sqrt{(1+e^x)^3}+C$$
Work Step by Step
$$A=\int e^x\sqrt{1+e^x}dx$$
Let $u=1+e^x$.
We would have $du=e^xdx$.
Also, $\sqrt{1+e^x}=\sqrt u=u^{1/2}$
Substitute into $A$, we have $$A=\int u^{1/2}du$$ $$A=\frac{u^{3/2}}{\frac{3}{2}}+C$$ $$A=\frac{2\sqrt{u^3}}{3}+C$$ $$A=\frac{2}{3}\sqrt{(1+e^x)^3}+C$$
