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Why can we say the following:

$u=\int\frac{\mathrm du}{\mathrm dx}\mathrm dx$,

where we treat $u$ as a function (e.g., see this proof).

Because as far as I know, $\int\frac{\mathrm du}{\mathrm dx}\mathrm dx$, as an indefinite integral, stands for a set of functions, whose derivative equals the integrand.

$\int f(x)\,\mathrm dx=F(x)+C$

So how can we say a function $u$ is equal to a family of functions?

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    To make a joke out of it: to avoid forgetting to put the constant every time, you just never write it. – 2017-01-27

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It is correctly written as $$ u(t) = \int_a^t \frac{du}{dx} \, dx $$ where $a$ is some chosen constant. You are correct, without the bounds, it is a family of functions. What you've written is an abuse of notation.

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    Wouldn’t the integral actually be the following? $$\int_a^t\frac{du}{dx}\,dx=u(x)\Bigr|_{x=a}^{t}=u(t)-u(a)$$ – 2017-10-25
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    @ChaseRyanTaylor That is correct. – 2017-10-25
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The antiderivate is never unique. If $f(x)$ has antiderivate $F(x)$, then $F(x)+c$, $c$ arbitary real, is an antiderivate as well.

The indefinite integral contains this constant and we can verify $F'(x)=f(x)$ no matter which value $c$ has. Therefore the indefinite integral can be considered to be the family of antiderivates.