How can this function have two different antiderivatives?

I’m currently operating with the following integral:

$$\int\frac{u'(t)}{(1-u(t))^2} dt$$

But I notice that

$$\frac{d}{dt} \frac{u(t)}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$

and

$$\frac{d}{dt} \frac{1}{1-u(t)} = \frac{u'(t)}{(1-u(t))^2}$$

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

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