# How can this function have two different antiderivatives?

I’m currently operating with the following integral:

But I notice that

and

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?