Interpret integral symbolically

Question

I am trying to calculate the integral of an symbolic function f[x] and want it to be symbolically $\int f(x) dx = F[x]$. I tried defining it implicitly via

F'[x_] = f[x];

This works fine in one direction as

D[F[b], b]

gives

f[b].

However, the otherway round doesn't work as the input Integrate[f[b], b] translates to $\int f(x) dx$ instead of F[b].

I also tried defining Integrate[f[x],x]=F[x], but this is protected.

Do you have any suggestions on how to get this working? I couldn't find anything in the documentation - I'm happy for hints in case I missed something.

Thanks a lot in advance.

Answer 1

Here is what you probably want. Associate UpValues with the symbols f and F as follows:

ClearAll[f, F]
f /: Integrate[f[x_], x_] := F[x]
F /: D[F[x_], x_] := f[x]

Then,

Integrate[f[t], t]
D[F[t], t]
(* F[t] *)
(* f[t] *)