Evaluating and Extending the Function ( I_k ) to Real-Valued ( b2 )

Consider the more general integral for integer ( k geq 3 ):

Introduction

In this article, we will delve into the derivation of the integral:

The ( I_k ) Integral

Specifically, we focus on the evaluation of:

[ I_k int_{0}^{infty} frac{x^k (e^{3x} - e^x)}{(e^x - 1)^4}, dx ]

This integral can be expressed as follows:

Integral Transformation and Differentiation

Notice that:

Using the Geometric Series

By leveraging the geometric series, we have:

[ frac{1}{1 - x^3} frac{1}{2} sum_{n0}^{infty} (n 1)(n 2)x^n ]

Thus, we can rewrite the integral ( I_k ) as:

Substitution and Simplification

Multiplying both the numerator and denominator of ( I_k ) by ( e^{-3x} ), we obtain:

[ I_k frac{partial^k}{partial a^k} int_{0}^{infty} frac{e^{ax} e^{-3x} (e^x - 1)}{(1 - e^{-x})^3} , dx Bigg|_{a1} ]

This further simplifies to:

[ I_k frac{1}{2} sum_{n0}^{infty} (n 1)(n 2) frac{partial^k}{partial a^k} int_{0}^{infty} e^{ax} e^{-3x} (e^x - 1) e^{-nx} , dx Bigg|_{a1} ]

Partial Derivative Evaluation

The partial derivative evaluation is straightforward:

[ frac{partial^k}{partial a^k} int_{0}^{infty} e^{ax} e^{-3x} (e^x - 1) e^{-nx} , dx Bigg|_{a1} frac{partial^k}{partial a^k} left[ frac{1}{(n 2 - a)} frac{1}{(n 3 - a)} right] Bigg|_{a1} ]

This simplifies to:

[ k! left[ frac{1}{(n 1)^{k - 1}} frac{1}{(n 2)^{k - 1}} right] ]

Summation of the Integral

Substituting back, we get:

[ I_k frac{k!}{2} sum_{n0}^{infty} (n 1)(n 2) left[ frac{1}{(n 1)^{k - 1}} frac{1}{(n 2)^{k - 1}} right] ]

This can be further simplified to:

[ I_k frac{k!}{2} sum_{n0}^{infty} left[ frac{(n 2)}{(n 1)^k} frac{(n 1)}{(n 2)^k} right] ]

Which results in:

[ I_k frac{k!}{2} left[ frac{2}{1^k} frac{4}{2^k} frac{6}{3^k} frac{8}{4^k} cdots right] k! left[ frac{1}{1^k} frac{2}{2^k} frac{3}{3^k} frac{4}{4^k} cdots right] k! zeta(k - 1) ]

Conclusion

For ( k 5 ), we obtain:

[ I_5 5! zeta(4) frac{4}{3} pi^4 ]

Where ( zeta(4) frac{pi^4}{90} ).

Extension to Real-Valued ( b2 )

Interestingly, WolframAlpha hints that a similar result can be extended to any real-valued ( b2 ):

[ int_{0}^{infty} frac{x^b (e^{3x} - e^x)}{(e^x - 1)^4} , dx Gamma(b 1) zeta(b - 1) ]

While a formal proof for this extension is left to the interested reader, this extension opens up further avenues for exploration and applications in mathematical analysis.