Need help with this difficult Integral involving exponentials

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In summary, the conversation discusses the integral \int \frac{x^2}{e^x-1}dx and whether it can be solved in closed form. The attempt at a solution involves using integration by parts, but the last integral \int x \ln{(e^x-1)} dx cannot be integrated. A series solution is proposed by expressing \frac{1}{e^x-1} as a series and subbing it into the original integral. However, the result cannot be expressed without the help of the error function. It is suggested that the integral may only be solvable for definite integrals over 0 to infinity.
  • #1
DiogenesTorch
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Homework Statement



This is the integral
[tex]
\int \frac{x^2}{e^x-1}dx
[/tex]


Homework Equations


Can this even be solved in closed form?


The Attempt at a Solution


Only method I can think of is integration by parts over numerous steps. I did that until I get the following:

[tex]
\int \frac{x^2}{e^x-1}dx = x^2 \ln{(e^x-1)} -\frac{5x^3}{3}-2 \int x \ln{(e^x-1)} dx
[/tex]

Now I am stuck on the last integral above: [itex]\int x \ln{(e^x-1)} dx[/itex] and can't find how to integrate it.

Any ideas or suggestions much appreciated :)
 
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  • #2
Is this acceptable?

Can we do this? If x is taken to always be positive then:

[tex]
\begin{align*}
\frac{1}{(e^x-1)} &= \frac{e^{-x}}{1-e^{-x}} \\
&= e^{-x}\frac{1}{1-e^{-x}} \\
&=e^{-x}\sum_{n=0}^{\infty}e^{-nx} \\
&=\sum_{n=0}^{\infty}e^{-(n+1)x} \\
&=\sum_{n=1}^{\infty}e^{-nx}
\end{align*}
[/tex]

Then subbing this series into the original integral

[tex]
\begin{align*}
\int \frac{x^2}{e^x-1}dx &= \int x^2 \Biggr( \sum_{n=0}^{\infty}e^{-(n+1)x} \Biggr) dx \\
&= \sum_{n=0}^{\infty} \int x^2 \Biggr( e^{-(n+1)x} \Biggr) dx \\
&= \sum_{n=1}^{\infty} \int x^2 e^{-nx} dx
\end{align*}
[/tex]
 
  • #3
You cannot solve it in closed form, you need the polylogarithm function. You can find series solutions though.
 
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  • #4
Thanks micromass,

So is the series solution above okay are am I totally off on it?
 
  • #5
DiogenesTorch said:
Thanks micromass,

So is the series solution above okay are am I totally off on it?

The series solution looks fine, I don't think you can express the result of final integral without the help of error function. Note that the result is valid when ##0<x<\infty##.

Are you sure you were asked for an indefinite integral instead of a definite one?
 
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  • #6
Pranav-Arora said:
The series solution looks fine, I don't think you can express the result of final integral without the help of error function. Note that the result is valid when ##0<x<\infty##.

Are you sure you were asked for an indefinite integral instead of a definite one?

Thanks Pranav.

Looking back this has to be only for definite integrals as ##x## was a change of variable for a frequency relationship, ##x=\frac{\hbar\omega}{k_b T}##. So it has to be only for definite integrals over ##0<x<\infty##.
 
  • #7
This integral has a solution if the boundary is from 0 to infinity but I do not know if this is relevant to you.
 

1. How do I solve an integral involving exponentials?

Solving integrals involving exponentials is a common problem in calculus. The key is to use the appropriate integration techniques, such as integration by parts or substitution, to simplify the integral. It may also be helpful to use the properties of exponentials, such as the power rule, to simplify the expression before integrating.

2. What are the steps to solving a difficult integral involving exponentials?

The first step is to identify which integration technique will be most useful for the given integral. Then, simplify the expression using properties of exponentials. Next, apply the chosen integration technique and solve the resulting integral. Finally, check your answer and simplify if necessary.

3. Can I use a calculator to solve an integral involving exponentials?

While a calculator may be helpful in checking your answer, it is important to understand the steps and techniques used to solve the integral. Calculators may also have limitations or errors, so it is best to rely on your own understanding and knowledge.

4. How can I check if my answer to an integral involving exponentials is correct?

One way to check your answer is to differentiate it and see if it results in the original function. Another method is to use a graphing calculator or software to graph both the original function and your integral, and see if they match.

5. Are there any tips for solving difficult integrals involving exponentials?

Some useful tips include simplifying the expression before integrating, using properties of exponentials, and breaking the integral into smaller parts. It can also be helpful to practice solving different types of integrals to become more familiar with the techniques and strategies involved.

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