# Real Analysis/Fundamental Theorem of Calculus

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The **Fundamental Theorem of Calculus** is said to be the central theorem of elementary calculus. It states effectively that "Differentiation" and "Integration" are inverse operations.

Conventionally, the theorem is presented in two parts

## First Form of the Fundamental Theorem[edit]

### Theorem[edit]

Let

Let be differentiable on and let for all

Let be Riemann integrable on

Then,

#### Proof[edit]

Let and let be given.

Then, there exists such that for a partition implies that

Consider a partition and let . By Lagrange's Mean Value Theorem, we have that there exists that satisfies

Let the tagged partition be the partition along with the tags

Thus,

But we know that and hence, .

As is arbitrary, that is,

## Second Form of the Fundamental Theorem[edit]

We first define what is known as the "Indefinite Integral"

### Definition[edit]

Let be Riemann integrable on .

We define the **Indefinite Integral** of to be the function given by

for all

### Theorem[edit]

Let be continuous at

Let be the indefinite integral of

Then, is differentiable at and

#### Proof[edit]

Let be given, and let but

Observe that (say).

There exists such that if a partition then, (note that in this proof, all the Riemann sums are over the interval ).

As is integrable over , it is bounded over that interval. Hence, let . Thus,

As is continuous at , there exists such that whenever .

Now consider

Then,

That is, , or