Is indefinite integration suspect? In this post, Qiaochu Yuan remarks that 'it is convenient but misleading to write
$$
\int f(x) \, dx=g(x)
$$
[where the derivative of $g$ is $f$]'. This sentiment seems to be shared by many contributors here, and I don't understand why. To me, both definite and indefinite integration are both valid operations you can perform on a function, and there is nothing suspect about indefinite integration.
I know about the fundamental theorem of calculus, which (as far as I understand) explains the link between indefinite and definite integration. If by integration we mean computing the area under the graph, the fundamental theorem of calculus shows us that integration is the opposite of differentiation, since
$$
\frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x)
$$
This shows that every continuous function has an antiderivative. Since a clear link between integration and antidifferentiation has been established, we give the antiderivative the convenient label 'indefinite integral'. (This also explains why the definite and indefinite integration notations are so similar.) This label is fine, so long as we remember that integration is defined as finding the area under the graph, while antidifferentiation is defined as finding the inverse of the derivative.
Another result of the fundamental theorem of calculus is that
$$
\int_{a}^{x}f(t) \, dt=\int f(x) \, dx
$$
So obviously every indefinite integral can be rewritten in terms of definite integrals, but I don't understand the motivation behind this. If $F$ is an antiderivative of $f$, then why is it more correct to write
$$
\int_{a}^{x} f(t) \, dt = F(x) \, ,
$$
compared to
$$
\int f(x) \, dx = F(x) \, ?
$$

If we define a function $F(x)$ as the antiderivate of the function $f(x)$ then by FTC we see that:
$$\int_a^xf(t)\,dt=F(x)-F(a)\neq F(x)\forall f(x)$$
The problem lies in that for a given known derivative, $F'(x)=f(x)$, there are many different equations which satisfy $F(x)$ which differ by a constant, often notated by${}+C$. This can be seen in many cases where people neglect this "constant of integration":
$$\int\frac{1}{ax}\,dx=\frac{1}{a}\int\frac 1x\,dx=\frac1a\ln|x|+C_1$$
$$\int\frac1{ax}\,dx=\frac1a\ln|ax|+C_2$$
which at first seem completely different but when the correct values of $C_1,C_2$ are chosen these are the same, see?

I think the point that the post is trying to make is that while in many scenarios it is easy to define a new function as the antiderivative of another, some may find this as misleading by thinking there is only one function $g(x)$ which satisfies these conditions, when in reality it is what we call a "family" of similar functions
