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May 26, 20260 citationsOpen Access

Proving that (x+1) ¹ x+1

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MAMustafa Al-Quzweeni

Key Points

  • The research aims to demonstrate the inequality (x+1)^1 ≠ x+1 and address the properties of integrals related to logarithmic functions and poles.
  • Proved that (x+1)^1 differs from x+1 through algebraic exploration.
  • Examined the integral of 1/x and its relationship to constants and poles during differentiation and integration.
  • Explored regularization methods for specific integrals and converging series via the zeta function.
  • Established that the integral of 1/x is not simply lnx due to omitted poles, altering integration outcomes.
  • Demonstrated converging series approximations, achieving high accuracy with series and integral transformations.
  • Found that the integral of 1/Γ(x) from -∞ to ∞ equals e, revealing deep connections between functions.

Abstract

I will be proving that (x+1) ¹ x+1, then I will prove that the integral of 1x is not lnx in fact is missing an entire pole, and this pole reverses the information lost when diffrentiating a constant, making the +C we add after integrating unecessary, and in the process will be revealing infinitely small terms that was hidden from our sights, that will allow me to show the proper expansion of (x+1) ¹, then I will be showing the regularization of diverging integrals for ₀^1xⁿ dx when n<-1 and a relation that converts ₀^f (x) dx into ₍=₁^f (n) using the mclaurin series and the zeta function, which could also be used to get high accuracy approximations like ₍=₁^1n (eⁿ-1) ²6-ln22-124, I will also show that -^1 (x) dx=e.

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Cite This Study

Mustafa Al-Quzweeni (2026) studied this question.

synapsesocial.com/papers/6a153b00b5d9c58d83e8d2e2https://doi.org/10.5281/zenodo.20360738
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