B (upper limit) ∫a (lower limit) f(x)dx=F(b)-F(a), and f(x) is the original function of F(x).

Proof: We have proved that φ' (x) = f (x), so φ (x)+c = f (x).

But φ (a) = 0 (the integral interval becomes [a, a], so the area is 0), so f (a) = C.

So φ (x)+f (a) = f (x), when x=b, φ (b) = f (b)-f (a),

And φ (b) = b (upper limit) ∫a (lower limit) f(t)dt, so b (upper limit) ∫a (lower limit) f(t)dt=F(b)-F(a).

Re-writing T as X becomes the formula at the beginning, which is Newton-Leibniz formula.