ABSTRACT
In our investigations we have used the following two versions of the fundamental theorem of calculus:
if f is differentiable then ∫ b
f ′(t) dt = f(b) − f(a) ; (1)
if f is continuous and a ≤ x ≤ b then d
dx
∫ x
f(t) dt = f(x). (2)
At first glance both are similar in showing that differentiation and integration are inverse operations. We have
∫ d
dt (f) =
d
dx
∫ f = f.