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I need helping computing the following indefinite integral $$\int\frac{e^x}{x}\mathrm dx$$

I tried integration by parts but did not yield any result.

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    $\begingroup$ It cannot be integrated, it's a special function. $\endgroup$
    – m0nhawk
    Feb 14, 2014 at 12:38
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    $\begingroup$ See exponential integral. $\endgroup$
    – Lucian
    Feb 14, 2014 at 12:39

2 Answers 2

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$\int\dfrac{e^x}{x}dx$

$=\int\dfrac{1}{x}\sum\limits_{n=0}^\infty\dfrac{x^n}{n!}dx$

$=\int\sum\limits_{n=0}^\infty\dfrac{x^{n-1}}{n!}dx$

$=\int\left(\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{x^{n-1}}{n!}\right)dx$

$=\ln x+\sum\limits_{n=1}^\infty\dfrac{x^n}{n!n}+C$

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The following answer comes from Symbolab and Wolfram Mathematica.

The common rule for integrating $\frac {e^x}{x}$ is: $$\int \frac{e^x}{x} dx = Ei(x) + C$$ $Ei(x)$ [if you didn't know] is what's called the Exponential Integral function, and is computed when you are raising Euler's Number to the power of a variable and dividing the exponent by the variable. According to Wolfram MathWorld, this function is closely related to the incomplete gamma function, which is this: $$\Gamma(0, x) = -Ei(-x) + \frac 12 [\ln(-x) - \ln(-\frac 1x)] - \ln x$$ Since you are trying to compute $\int \frac {e^x}{x} dx$, your answer is $Ei(x)$.

References:

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