$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$ If $f(x)$ is a continuous function on $\mathbb{R}$ and I am asked to find 
$\int f(x) \, dx$, what is the problem with the following answer:

$$\int f(x)\,\mathrm{d}x = \left(\int_0^x f(t) \, \mathrm{d}t\right) + C$$
The problem with your answer is the problem with your answer.


And I hope the problem with my answer will tell you something about the problem with your answer.
There are cases when it's appropriate do this:  

Someone says


  $e^{-x^2}$ is not integrable  


So we say


  Of course it is integrable.  All continuous functions are integrable.  The integral is $\int_0^x e^{-t^2}\;dt$.  What you mean is: the integral is not an elementary function.