TITLE: Integrate $\int e ^ \sqrt{x} \, dx$
QUESTION [8 upvotes]: $$\int e ^ \sqrt{x} \, dx$$
I don't even know how to begin. I tried u sub but obviously doesn't work, no variable to cancel. I tried the formula to memorize $$\int ae^ {au} \, du = \frac{1}{a^2} (au - 1)e^au$$ but that didn't work and I think maybe it isn't suppose to.

REPLY [5 votes]: Let's put $x = y^2,\;\;\sqrt x = y.\;\;$ Then $\,dx = 2y\,dy$. 
Substituting, this gives us:
$$\int e^{\sqrt{x}}dx=2\int ye^ydy$$
which lends itself nicely to integration by parts.
$$\begin{align} 2\int ye^y\,dy & = 2\left( ye^y - \int e^y\,dy\right) \\ \\
& = 2\left(ye^y - e^y\right) + C \\ \\ 
& = 2e^y\left(y - 1)\right) + C \\ \\ 
& = 2e^{\sqrt x}\left(\sqrt x - 1\right) + C \end{align}$$