Chapter 5 - Accumulating Change: Limits of Sums and the Definite Integral - 5.9 Activities - Page 409: 6

$\dfrac {2}{3}\sqrt {\left( 1+e^x\right) ^{3}}+C$

$u=1+e^{x}\Rightarrow du=e^{x}dx$ $\Rightarrow \int e^x\sqrt {1+e^x}dx=\int \sqrt {1+e^x}e^{x}dx=\int \sqrt {u}du=\int u^{\frac {1}{2}}du=\dfrac {2}{3}u^{\frac {3}{2}}+C=\dfrac {2}{3}\sqrt {\left( 1+e^x\right) ^{3}}+C$