How can I find the definite solution for this integral using Mathematica?

[rabbahs]

Homework Statement

Integration of following is required, I have tried on Mathematica but it didnt works, may be some special function will be available to find its definite solution. All variable and known data is presented in the following equation.

Homework Equations

The Attempt at a Solution

when i tried to solve it in Mathematica, it give me the following msg,

 

[gomunkul51]

I'll try explain what you got from the program:

in your question the integral is from u to infinity and you tried to calculate it in Mathematica as an integral from 0[/] to infinity.

your integral diverges i.e. the sum that the integral calculates goes to infinity near 0.
in other words it has a singularity there.

but if you can try to calculate in some ways, tell us in what course you got it (what tools you've learned) and may be we point you to the right way :)

*could be represented as a series and could be split into several parts one of one of them the integral is just the Exponential Integral or Ei...

Good Luck

 

[rabbahs]

thanks gomunkul51,

you are right about the wrong discription of Integral into Mathematica.
Now when i put the correct limits, its gives me the following answer. which is surely what I don't want.

This is not a part of any course work. During my research project I am stuck to find the solution of that integral. Still now my hands are tight.

What I think is that if is a kind of special function.
some thing I found on net which i related to the solution. Kindly view the following,

but the problem is that I also expand in in integral (as u can see the far most side of the equation.

Then in a research paper I found a solution of this function,

The reference to about solution is "Rainville, E.D., Special Functions, 44 (Macmillan, 1960)."

And i don't have any access to that reference.

If i cannot justify the solution then it is hard to me to use it.

any other idea ? most welcome

 

[phyzguy]

I don't understand your issue. Mathematica has given you the correct answer. Presumably u is a real number greater than 0, otherwise the integral doesn't converge. Mathematica says that if this is the case, then the answer is \frac{e^-u}{u}+Ei(-u) The exponential integral function is a tabulated function which you can look up. Mathematica also has the Ei function built in so you can plot it or get the numerical values. What is wrong with this solution? Why is it "surely what you don't want?

As an aside, you really should use different variables for the integration variable and the limit variable, like this:

Integrate[Exp[-v]/v^2, {v, u, \[Infinity]}]

I'm surprised Mathematica didn't get confused by this, but it didn't.

 

[gomunkul51]

If it's not part of a class that you need to answer is a "particular" way, then basically you get what you've got:

original integral = (1/x*e^x) - Ei(1,x)

Ei is a known function which defined as: integral[-infinity,x]((e^u)/u)