1. ## natural exponential integral

$\int e^{-\sqrt{x}}$

Surely, this must be on this forum somewhere?

2. You could use integration by parts.

Let $u=e^{-\sqrt{x}}, \;\ dv=dx, \;\ v=x, \;\ du=\frac{-1}{2e^{\sqrt{x}}\sqrt{x}}dx$

3. here goes.

$u = \sqrt x$

$du = \frac {1}{2} \frac {1}{ \sqrt x} dx$

$du = \frac {1}{2} \frac {1}{u} dx$

$dx = 2u du$

so we get

$2 \int u e^{-u} du
$

use integration by parts, to finish this off.

I get $-2 \sqrt x e^{-\sqrt x} - 2 e^{-\sqrt x } + C$