To calculate the expectation of $e^x$ for a standard normal distribution I eventually get, via exponential simplification:
When I plug this into Wolfram Alpha I get $\sqrt e$ as the result. I'd like to know the integration step(s) or other means I could use to obtain this result on my own from the point I stopped. I am assuming that Wolfram Alpha "knew" an analytical solution since it presents $\sqrt e$ as a solution as well as the numerical value of $\sqrt e $. Thanks in advance!