This gives us $du = e^{x} dx$. Substituting these into the integral, we get
\[
\int \sqrt{u} du
\]
which can be solved to give
\[
\frac{2}{3} u^{\frac{3}{2}} + C
\]
Substituting $u = e^{x} - 1$ back in, we get
\[
\frac{2}{3} (e^{x} - 1)^{\frac{3}{2}} + C
\]
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