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 $$ \int f(x) \, dx = F(x) \, ? $$