Q:

Does $\int e^\frac 1x \, \mathrm dx$ has a closed form? 

How can i solve the integral $\int e^\frac 1x \, \mathrm dx$? 
I came across this one while trying to do multiple integral on $\int\int_D e^{(\frac xy)} \, \mathrm dA$ where D is the area between $y=1$ and $y=\sqrt x$
I know i solve this question by doing $\int^1_0 dy\int^{y^2}_0(e^{\frac xy}dx)$. But i don't like 'skipping' a possible solution , Any guidelines? 

A:

The integral is not elementary.   But Maple provides a closed form in terms of an "exponential integral" function:
$$
\int \!{{\rm e}^{{z}^{-1}}}{dz}=z{{\rm e}^{{z}^{-1}}}+{\rm Ei}_1 \left(-{z}^{-1} \right)
$$
The definition:
$$
{\rm Ei}_a \left(z \right) =\int _{1}^{\infty }\!{{\rm e}^{-tz}}{t}^{
-a}{dt}
$$

