In an article i recently read (The American Put Option and Its Critical Stock Price by David S. Bunch and Herb Johnson) the authors presented this formula as something very general and as common knowledge

$$P = \mathop {\max }\limits_{{S_c}} \int\limits_0^T {{e^{ - rt}}(X - S_c)} fdt,\quad (S > {S_C})$$ 


where P, r, T, X, and Sc are the American put price, risk-free rate,
time to maturity, exercise price, and critical stock price,
respectively.Let S be the current stock price (at time  t= 0).f,
is the first-passage probability,

However i cant recall that i have seen this formula AND f in the same formula, what am I missing? Where did this formula come from? How would one go about if one needs to derive this formula? Is there a similar formula ?

I haven't read that paper in a while and I don't see an online accessible link. But I strongly suspect it is supposed to be some kind of approximation. It doesn't look correct if meant to be exact. For example, letting r -> 0, P should tend to the euro-style value. But since Sc->0 for t < T, f would also tend to 0 for t<T and that formula would say P->0, not P -> P_euro. 

this is how it is presented in the article,

Maybe: maybe you are thinking $S_c$ is a number?  It is a curve. From that paper, "The maximization is taken over all possible functions $S_c(\tau)$, where $\tau \equiv T-t$."  For the limit Alan is testing, the curve $S_c(\tau)$ is approximately zero for $\tau > 0$ but goes to the strike at expiry.  That curve is hit at expiry or not at all, giving exercise at expiry if the option is in the money, i.e. the European option limit.

Maybe: maybe you are thinking $S_c$ is a number?  It is a curve. From that paper, "The maximization is taken over all possible functions $S_c(\tau)$, where $\tau \equiv T-t$."  For the limit Alan is testing, the curve $S_c(\tau)$ is approximately zero for $\tau > 0$ but goes to the strike at expiry.  That curve is hit at expiry or not at all, giving exercise at expiry if the option is in the money, i.e. the European option limit.

Actually, I knew that before , I was after where it came from, like origin since i have not seen this with $f$ included anywhere else .