Integration of the function

[subharpe]

Hi everybody, someone can give me the integration technique for the function
∫exp(x)*ln(x) dx? Somebody todl me it is not integrable. I want to know how is it non-integrable and if not, how to integrate it?
One way is to expand exp(x) and integrating. But I couldn't find a concrete result, the result comes in a series not in analytic expression.

 

[subharpe]

Err...................
Someone please reply.........

 

[dmk]

https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions

Added after 5 minutes:

So,

 

[subharpe]

Thnx for the link, but how to derive ? I searched for what Ei(x) means but it is also a infinite series whose value at all x are not known. Is there any way to recognize by looking at it which function can be integrated upto a compact expression and which function is not? Like elliptical integral, gaussian distribution, normal distribution, error function all are defined in terms of some integral but they cannot be simplified to an algebric expression.

 

[dmk]

I am afraid you must realize a series.

 

[jcy]

I think you can derive it simply be integrating by parts.

let
u=e^x v = ln|x|
du = e^x dx dv = dx/x

∫e^x ln(x) dx = uv - ∫udv = e^x ln|x| - ∫e^x/x dx

The last integral is recognized as the exponential integral. Although you cannot integrate it analytically, you can find good approximations for it in many references (such as Abramowitz and Stegun, chapter 5). The convergence depends on your limits of integration however.