Answer
$2\sqrt{e^x+1}+c $
Work Step by Step
Recall that $(e^x)'=e^x$
Let $ u=e^x $, then $ du=e^x dx $ and hence we have
$$
\int \frac{e^x}{ \sqrt {e^x+1}} dx=\int \frac{1}{ \sqrt {u+1}} du\\
=\int (u+1)^{-1/2}du=\frac{1}{1/2} (u+1)^{1/2}+c=2\sqrt{e^x+1}+c
.
$$
