# Answer on Differential Equations Question for Arieta Tekaura

Question #4008

Integrate W.R.T x:

e^x(1+1/x)

Explain with steps please

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">

Need a fast expert's response?

Submit orderand get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

## Comments

Assignment Expert27.03.2012 09:58The 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 Tekaura19.08.2011 23:38The 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?

## Leave a comment