How can this function have two different antiderivatives?

I’m currently operating with the following integral:

u(t)(1u(t))2dt

But I notice that

ddtu(t)1u(t)=u(t)(1u(t))2

and

ddt11u(t)=u(t)(1u(t))2

It seems that both solutions are possible, but that seems to contradict the uniqueness of Riemann’s Integral.

So the questions are:

  1. Which one of them is the correct integral?
  2. If both are correct, why the solution is not unique?
  3. The pole at u(t)=1 has something to say?

Answer

It is not really a contradiction, since difference of the two functions is constant:
11u(t)u(t)1u(t)=1u(t)1u(t)=1.
(Derivative of a constant function is zero. Primitive function is determined uniquely up to a constant.)

Attribution
Source : Link , Question Author : Dargor , Answer Author : Martin Sleziak

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