# Use of Exponential Integral in solving an integral

If I'm solving a version of the following integral

$\int \log(x)e^x dx$

how does the exponential integral arise as a solution to this? (Or is it much too theoretical to explain simply?) I do a much more complicated version of this on WolframAlpha.com and the Ei(x) function is used, but it is used with the above integral too, and no steps are given.

I understand the form of Ei(x), but thought that someone might be able to explain how it arises as a solution to the above.

Thank you!

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Using integration by parts $$\int \log(x)e^x dx=\int \log(x)(e^x)^{\prime} dx=\log(x)e^x -\int (\log(x))^{\prime}e^x dx=\log(x)e^x -\int \frac{e^x}{x} dx$$ The last integral is the exponential integral $Ei(x)$
$\int\log{x}\ e^x dx=\log{x}\ e^x - \int\frac{e^x}{x}dx$ - integration by parts