Proving that (x+1)¹ ≠ x+1

I will be proving that (x+1)¹≠ x+1, then I will prove that the integral of 1/x 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 ∫₀¹xⁿ dx when $n<-1$ and a relation that converts ∫₀∞f(x)dx into ∑ₙ₌₁∞f(n) using the mclaurin series and the zeta function, which could also be used to get high accuracy approximations like ∑ₙ₌₁∞1/n(eⁿ-1)≈ π²/6-ln2π/2-1/24, I will also show that ∫-∞∞1/Γ(x)dx=e.