Definite Integral - Question about notation For an arbitrary function $f(t)$ we define:
\begin{equation}
\bar{f}(t) =\int^{t}_{0} f(\tau) d\tau
\end{equation} 
Is it true the following? 
For an arbitrary function $f(x,t)$ we define:
\begin{equation}
\bar{f}(x,t) =\int^{t}_{0} f(x) dx
\end{equation}
I get confused because it's a single integration but 2D function. 

No, the right-hand-side does not depend on x, since you integrate over it. 
You might define a function like 
$$\bar{f}(x,t)=\int_0^t f(x,y)dy$$
