The Fundamental Theorem of Calculus

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.