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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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2 Answers

up vote 1 down vote accepted

Using integration by parts \begin{equation}\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\end{equation} The last integral is the exponential integral $Ei(x)$

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Oh, very nice! I love it when things are simple. Thank you! –  SingularityFuture Dec 4 '12 at 19:08
    
You are welcome –  Nameless Dec 4 '12 at 19:13
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$\int\log{x}\ e^x dx=\log{x}\ e^x - \int\frac{e^x}{x}dx $ - integration by parts

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