If you just want to calculate the value of the integral, this should be fine if I'm not mistaken, as you might also call the upper bound $u$ and later set $u=x$ again after calculation. More precisely, to evaluate the integral, I'm parametrizing the upper bound to $u$ s.t. we consider
Thus, you may evaluate $\int_0^uf(t)dt$ in dependence on $u$ and then derive the value of the original integral via $h(x)$.
However, if you are working with the by the integral induced function
and want to perform more complex tasks like differentiation or integration of $g$, then this is called a parametric integral and the answer is no. But in this case, $x$ would also less be considered a constant value than a variable(for this respective function).