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I want to integrate $\int \frac{e^x}{x}dx$.

Attempt 1: Let $u = e^x$, and $dv = \frac{1}{x}dx$ i.e. $v = \log x$. Then by integration by parts, we have $$ \begin{align} \int \frac{e^x}{x}dx &= \int u\, dv \\ &= uv - \int v\, du \\ &= e^x\log x - \underbrace{\int e^x\log x\,dx}_{=?} \end{align} $$

Attempt 2: Let $u = \frac{1}{x}$ and $dv = e^xdx$. Then similarly we have $$ \begin{align} \int \frac{e^x}{x}dx &= uv - \int v\, du \\ &=\frac{e^x}{x} - \int e^x\Big(-\frac{1}{x^2}\Big)dx \\ &=\frac{e^x}{x} + \underbrace{\int\frac{e^x}{x^2}dx}_{=?} \end{align} $$

I tried choosing different $u$s by using integration by parts, but none worked

Does anyone have a better idea?

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  • $\begingroup$ This is not an elementary integral. Check the exponential integral. $\endgroup$ – Mhenni Benghorbal Jan 6 '15 at 8:05
  • $\begingroup$ I don't think this is a very simple integral to evaluate, but what you have done is correct. $\endgroup$ – Bman72 Jan 6 '15 at 8:05
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    $\begingroup$ It would be nice if you would try to learn mathjax, see help page, and not to expect us to read your not too excellent handwriting. $\endgroup$ – Karl Jan 6 '15 at 8:34
  • $\begingroup$ sorry about my awkward handwriting ,as my native language is not English , it is not easy for me to surf through the whole website , I will try using what you called mathjax next time ,thank you $\endgroup$ – user143997 Jan 6 '15 at 8:55
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    $\begingroup$ @user143997 here is a link that you can use to learn the fundamental of mathjax (/latex) $\endgroup$ – Bman72 Jan 6 '15 at 10:11
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This is a textbook example of an integral that can't be expressed with traditional set of analytical functions. With a specific choice of the integrating constant, you get the exponential integral $\operatorname{Ei}(x)$.

http://en.wikipedia.org/wiki/Exponential_integral

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  • $\begingroup$ thank you very much ,orion ,now I see what I am gonna do $\endgroup$ – user143997 Jan 6 '15 at 8:56

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