65 728
Assignments Done
99,3%
Successfully Done
In October 2018

# Answer to Question #4008 in Differential Equations for Arieta Tekaura

Question #4008
Integrate W.R.T x:
e^x(1+1/x)
<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">

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Assignment Expert
27.03.12, 16:58

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.

Arieta Tekaura
20.08.11, 06:38

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?