Last term, in my diff eq class during an exam and my professor made a mistake on one of the problems :rolleyes:. Anyway, the problem he gave us was pretty difficult to integrate.

Integrate with respect to x: e^x/x

Substitution and by parts didn't seem to work. The closest I got was to separate the expression as

(1/x)(sum from n=1 to infinity of x^n/n!), I believe that is the correct expansion of e^x, I may be wrong.

I'm just wondering if there is another approach to this problem.

Yes, this is a toughy. It is not soable by elementary means. That is why parts and

substitution would not work. Your professor probably meant \int xe^{x}dx

\int\frac{e^{x}}{x}dx

This has no closed form in elementary terms.

Over the yesrs mathematicians have came up with forms in order to represent things like this that are hard to deal with.

This one of yours is a form of the Exponential Integral, Ei(x), and can be represented by

-Ei(1,-x)

If you had limits, then we could probably evaluate. But an indefinite form is another matter.

Actually he meant x/e^x, which can be integrated by parts.

But thank you non-the-less, I think I'm going to read up some more about this Ei(x) form!

Just Google the Exponential Integral. It is beyond elementary calculus and can be found in suh texts as Mathematical Physics.

Ei(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt

One can find the asymptotic series for the expoenntial integral or express it as an Incomplete Gamma function.