How can I solve this integration by parts problem for the function x^2/(e^x+1)?

[Mentallic]

Homework Statement

$$I=\int{\frac{x^2}{e^x+1}dx}$$

The Attempt at a Solution

I tried integration by parts but that didn't work because it just became more complicated in the end.

$$I=x^2ln(e^x+1)-2\int{xln(e^x+1)dx}$$

Then, $$\int{xln(e^x+1)dx}=xln(e^x+1)-\int{\frac{x}{e^x+1}dx}$$

It doesn't seem to be going anywhere and is only getting longer and more confusing.

Any ideas?

 

[gabbagabbahey]

You seem to be saying that $\int\frac{1}{e^x+1}dx=\ln(e^x+1)$...have you tried double checking that by taking the derivative?

 

[Cyosis]

Is this a homework problem or a problem you have made up yourself? Secondly is it supposed to be an indefinite integral or are there upper and lower limits in the problem statement in your book?
The reason why I am asking these questions is that you cannot solve the indefinite integral in terms of elementary function.

 

[Mentallic]

You seem to be saying that $\int\frac{1}{e^x+1}dx=\ln(e^x+1)$...have you tried double checking that by taking the derivative?

Oh yeah oops

Is this a homework problem or a problem you have made up yourself? Secondly is it supposed to be an indefinite integral or are there upper and lower limits in the problem statement in your book?
The reason why I am asking these questions is that you cannot solve the indefinite integral in terms of elementary function.

This question was from a secondary source. i.e. I was reading responses to a trial maths exam and someone posted this question, asking how it could be done, and I realized I couldn't answer it either.

There may be upper and lower limits, but what difference does this make if it cannot be expressed in terms of elementary functions? Also, is there a simple reason as to how you knew it could not be expressed simply? Maybe by experience?

 

[Cyosis]

I knew that it couldn't be expressed in terms of elementary functions because it's a fairly common integral in statistical mechanics. In general it's quite hard to find out if an integral cannot be computed. Generally people will know it through experience and what not.
That said it does matter if there are limits or not. For example if the limits would be from 0 to infinity you could compute the integral and get an exact answer.