(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Some friend of mine found this on a book:

[tex]\int_{0}^{+inf}\frac{1-cos(\omega t)}{e^{\omega /C}(e^{\omega /T}-1)\omega }=ln[\frac{(\frac{T}{C})!}{|(\frac{T}{C}-iTt)!|}][/tex]

The proof is left for the reader.

2. Relevant equations

3. The attempt at a solution

First very safe step:

cos(ωt)=Re(e^(iωt))

Second.1: A possibility is using a substitution of:x=e^ωBut now instead of1/ωwe have1/ln(x)which is difficult to handle in integration.

Second.2: I tried using derivation under the integral sign which I've basically never used before, assuming the legality of my move. Ifit=A.

[tex]I=\int_{0}^{+inf}\frac{1-e^{it\omega}}{e^{\omega /C}(e^{\omega /T}-1)\omega }=

\int_{0}^{+inf}\frac{1-e^{A\omega}}{e^{\omega /C}(e^{\omega /T}-1)\omega }[/tex]

[tex]\frac{d}{dA}I=

\int_{0}^{+inf}\frac{\partial }{\partial A}\frac{1-e^{A\omega}}{e^{\omega /C}(e^{\omega /T}-1)\omega }=\int_{0}^{+inf}\frac{-e^{A\omega}}{e^{\omega /C}(e^{\omega /T}-1)}[/tex]

Which may not be convergent. At least asωgoes to zero the function goes to infinite. And as it goes to infinite1/Tmust be greater than the exponent of the upper part of the fraction. And I still have to integrate with respect toAafter.

Alternative methods:

Use of representation by series. Maybe with the help of integration by parts.

Assuming the result is similar to the derivative result and just try differentiating.

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# Homework Help: Proof of already solved Hard Improper Definite Integral

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