Integrating e^x /x using Laurent series

[pivoxa15]

How does one integrate $$\int_{}^{} \frac{e^x}{x}dx$$

I could expand it using a Laurent series and than integrating term by term but are there more elementary methods?

 

[Stu091]

why can't you just use integration by parts twice? $$\int e^{x}\frac{1}{x} dx$$, $$u = e^{x}, du = e^{x}dx, dv = \frac{1}{x}, v = \ln x$$

This just keeps expanding. Using ILATE, the algebraic part should be u, but this just keeping increasing in the power on the bottom. I don't think it's a good candidate for parts.

Edit: I'm not sure what happened to that post. Disregard I guess.

 

[dextercioby]

Since

$$\int \frac{e^{x}}{x} \ dx =\mbox{Ei}\left(x)$$

you can search Abramowitz & Stegun's book on series expansion for this function.

Daniel.