The fundamental theorem of calculus is a like between the concept of the derivative of a function and the concept of integral. There are two parts to the theorem.

## First part

*This part is also known as the first fundamental theorem of calculus.*

## The theorem

Let f be a continuous real-valued function which is continuous on the closed interval [a,b]. Let F be the function defined, for all x in [a,b], by the following equation:

Then, F is a primitive of f on [a,b].

## Prerequisite information

- Have a knowledge of the definition of the primitive.
- Know the definition of the derivative.
- Know the mean value theorem.

## The proof

We want to find the primitive of F, denoted F′. This is why we use the defintion of the derivative to get started:

We know that f is continuous on [a,b], this is why it is continuous on [x,x+h] [x;x+h]⊂[a;b]. From the mean value theorem, we know that there exist a point c∈[x,x+h] such that:

Then, we deduce from the results (1) and (2) that:

When h→0, c→x, this is why we have:

## Second part

*This part is also known as the second fundamental theorem of calculus. Sometimes it is referred to as the evaluation theorem (in context of integral calculus).*

## The theorem

Let F be a primitive of f on [a,b]. If f is continuous on the interval [a,b], then:

## Prerequisite information

- Know the first part of the fundamental theorem of calculus.

## The proof

From the first part of the fundamental theorem of calculus, we have: