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Answer to Question #4008 in Differential Equations for Arieta Tekaura
Question #4008
Integrate W.R.T x:
e^x(1+1/x)
Explain with steps please
Expert's answer
<img src="/cgi-bin/mimetex.cgi?\int e^x (1 + \frac{1}{x})dx = \int e^xdx + \int \frac{e^x dx}{x} = e^x + Ei(x), \ where \ Ei(x)\ is \the \ exponential \ integral" title="\int e^x (1 + \frac{1}{x})dx = \int e^xdx + \int \frac{e^x dx}{x} = e^x + Ei(x), \ where \ Ei(x)\ is \the \ exponential \ integral">
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#233029 on May 2019
#233029 on May 2019 
Comments
The exponential integral is a special function defined on the complex plane. For real, nonzero values of x, the exponential integral Ei(x) can be defined as
Ei(x) = int(from -∞ to x) e^t/t dx.
The function is given as a special function because int e^t/t dx is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value, due to the singularity in the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and ∞. In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation elsewhere on the complex plane.
The answer is good, except for the last part. What is Ei(x)? How to work it out? Is it a constant or something else? Can you explain what is the exponential integral?
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