Fundamental theorem of calculus

The fundamental theorem of calculus part I

The first fundamental theorem of calculus is stated as follows:

Let f(t) be continuous function on the interval [a,b]. Then the function A(x) defined by the formula:

for all x in [a,b] is an antiderivative of f(x). That means

for all x in [a,b].

The fundamental theorem of calculus part II

Let F(x) be any antiderivative of f(s) on [a,b], so that F (x) = f(x) for all x in [a,b].

Then

.