IBP Struggles: Solving Integrals of \frac{x^2}{e^x+1} \& \frac{x^3}{e^x+1}

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In summary, the definite integral of a function with a particular constant term in the exponential can be evaluated by using the integration formula for a function with a different exponent.
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Homework Statement



Find the Integrals of [tex] \frac{x^2}{e^x+1}\\ \frac{x^3}{e^x+1} [/tex]


Homework Equations



Integration by parts

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.
 
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  • #2
We had a similar thread not too long ago, have a look here.
 
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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:

$$\int_0^{\infty} \frac{x^{s-1}}{e^x+1}\,dx=\left(1-2^{1-s}\right)\zeta(s)\Gamma(s)$$
 
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  • #4
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.
 
  • #6
Pranav-Arora said:
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:

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

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

Is there some way to get a new variation of this formula that can take these constants into account?
 
  • #7
Change variables to ##y = x/T##.
 
  • #8
Orodruin said:
Change variables to ##y = x/T##.

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

1. What is the general strategy for solving integrals of the form f(x)/g(x)?

The general strategy for solving integrals of the form f(x)/g(x) is to use a technique called integration by parts. This involves breaking the integral into two parts, one of which can be easily integrated while the other is differentiated. The resulting integral can then be solved using substitution or other techniques.

2. How do you choose which part of the integral to differentiate and which to integrate?

The choice of which part of the integral to differentiate and which to integrate is largely based on trial and error. However, a general rule of thumb is to choose the part that is more complicated or contains a higher degree polynomial to differentiate, and the part that is simpler or can be easily integrated to integrate.

3. What is the specific approach for solving the integrals x2/(ex+1) and x3/(ex+1) using integration by parts?

The specific approach for solving these integrals using integration by parts is as follows:

  • Let u = xn and dv = 1/(ex+1)
  • Use the formula ∫u dv = uv - ∫v du to find the integral
  • Solve the resulting integral using substitution or other techniques

4. Are there any special cases or tricks for solving these integrals?

Yes, there are a few special cases and tricks that can make solving these integrals easier:

  • If the integral contains a term in the form ex/(ex+1), it can be simplified to 1 - 1/(ex+1)
  • If the integral contains a term in the form x/(ex+1), it can be rewritten using the substitution u = ex

5. What are some common mistakes to avoid when solving these integrals?

Some common mistakes to avoid when solving these integrals include:

  • Misapplying the integration by parts formula
  • Forgetting to include the constant of integration
  • Making algebraic errors while simplifying the integral
  • Forgetting to use substitution or other techniques to solve the resulting integral

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