Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$
By the integral definition, g(x) is the area under the curve of f(x) from 1 to x.
eg: g(5) is the area under f(x) from 1 to 5.
I want to make the distinction that g(x) is NOT defined as an indefinite integral like $$g(x)=\int f(t)dt$$
Now, let's say I am given some arbitrary graph of f.
With the first definition (definite integrals), is g(x) still considered the anti-derivative of f(x)? This matters when exploring ideas about concavity and increase/decrease, and needing to examine the derivatives of g(x). Based on the F.T.C., I know that the derivative of the integral is the function itself. Can we say g'(x) = f(x) and that g''(x) = f'(x)? I know I can say this if g(x) was defined the 2nd way, with indefinite integrals.
But, can I say the same thing when given the limits of integration of $\int_1^x f(t)dt$ ? Or, is g(x) now specifically defined strictly as the area under the curve of f(x) from 1 to x? The 2 different types of integrals (definite vs indefinite) seem to mean different things. Only the 2nd one seems to fit into the antiderivative model (vs. being a specifically defined function.) I guess I am not seeing the connection b/w the two. Let me know if I am unclear.