Difference between indefinite and definite integration: $\int f(x) \;dx = \int^x_0 f(x) \; dx$? $\int f(x) \;dx = \int^x_0 f(x) \; dx$

I sometimes get the impression that there's meant to be a basic and important difference between definite and indefinite integration. But when I consider simple examples, it seems to me like the equation in the title holds, which makes it seem like there's not much difference.
 A: Definite integration gives, in principle, a definite real number back (if the upper bound happens to contain a variable, you get a function, but still).
Indefinite integration is usually interpreted to mean antidifferentiation, through the power of the fundamental theorem of calculus. It gives back a whole family of functions, with an unknown / generic constant term.
So in theory, these are quite different operations, and it's only through the miracle of FtC that they are so closely related (and thus get very similar notations). I have even heard people go so far as to say there is no such thing as indefinite integrals, there are only definite integrals and antiderivatives.
At any rate they are most definitely not equal. Best case, $\int_0^x f(t)\,dt$ (please don't use $x$ both as integration bound and integration variable) is one of the many $\int f(x)\,dx$.
A: Suppose $a$ is a value for which the definite integral $\int_a^xf(t)dt$ is well-defined; it is then a function of $x$. You can't write $\int_a^x f(x) dx$, because $x$ is the argument of this function, while $t$ is a dummy variable. You must write $\int_a^xf(t)dt$ instead of $\int_a^xf(x)dx$, just as one writes $\sum_{k=a}^xf(k)$ instead of $\sum_{x=a}^xf(x)$.
For such an $a$, $\int f(x)dx$ is shorthand for the set $\left\{C+\int_a^xf(t)dt|C\in S\right\}$ of functions (where $S$ is a number system of interest, such as $\Bbb R$ or $\Bbb C$), or (if the context permits) an arbitrary function in this set if we don't care which one we're talking about.
A: *

*The definite integral of $f(x)$ is a NUMBER and represents
the area under the curve f(x) from $x=a$ to $x=b$.
A definite integral has upper and lower limits on the integrals, and
it’s called definite because, at the end of the problem, we have a
number – it is a definite answer.

*The indefinite integral of $f(x)$ is a FUNCTION.
Indefinite integral usually gives a general solution to the
differential equation.
The indefinite integral is more of a general form of integration,
and it can be interpreted as the anti-derivative of the considered
function.

Fundamental Theorem of Calculus
The definite and the indefinite integral are linked by the Fundamental Theorem of Calculus as follows: In order to compute a definite integral, find the indefinite integral (also known as the anti-derivative) of the function and evaluate at the endpoints x=a and x=b.
The difference between definite and indefinite integrals will be evident once we evaluate the integrals for the same function.
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