# Hard Integral

1. Jun 17, 2014

### Eats Dirt

1. The problem statement, all variables and given/known data

Find the Integrals of $$\frac{x^2}{e^x+1}\\ \frac{x^3}{e^x+1}$$

2. Relevant equations

Integration by parts

3. The attempt at a solution

I did IBP twice and it seemed to just get bigger and uglier and now I am stuck. I found the solutions online of the integrals but still do not know how to do them myself.

2. Jun 17, 2014

### verty

We had a similar thread not too long ago, have a look here.

3. Jun 18, 2014

### Saitama

As said in the other thread, you need polylogarithms to express the final result. Instead, the definite integral from 0 to infinity can be easily evaluated and you have the following general result:

$$\int_0^{\infty} \frac{x^{s-1}}{e^x+1}\,dx=\left(1-2^{1-s}\right)\zeta(s)\Gamma(s)$$

4. Jul 1, 2014

### Eats Dirt

Hey, sorry for the atrociously late reply. I am not familiar with the two functions in the general form you posted. I recognize the one as a gamma function but have never seen the other.

5. Jul 1, 2014

### Orodruin

Staff Emeritus
6. Jul 1, 2014

### Eats Dirt

I have another integral where the term in the exponential is divided by a constant T so it takes the form of $$\frac{x^2}{e^\frac{x}{T}+1}\\ \frac{x^3}{e^\frac{x}{T}+1}$$

Is there some way to get a new variation of this formula that can take these constants into account?

7. Jul 1, 2014

### Orodruin

Staff Emeritus
Change variables to $y = x/T$.

8. Jul 1, 2014

### Eats Dirt

Then we just use the same integration formula from pranav? Is this because the limits of integration involve an infinity so any constant T in the denominator will have no effect?

9. Jul 1, 2014

### Orodruin

Staff Emeritus
It will have an effect, namely multiplying the result by a factor $T^{n+1}$, where n is the exponent of x, since
$$x^n\,dx = T^{n+1} y^n\,dy$$