Do you want to learn how to derive integral functions applying the **fundamental theorem of calculation**? I explain it to you step by step in this lesson, with solved exercises.

The first fundamental theorem of calculation tells us that integration is the inverse operation to derivation.

Índice de Contenidos

- 1 Formula of the fundamental theorem of the integral calculation
- 2 Solved exercises of the fundamental theorem of the calculation
- 3 Fundamental theorem formula to derive integrals ranging from a constant to a function
- 4 Fundamental theorem formula to derive integrals ranging from one function to another function

## Formula of the fundamental theorem of the integral calculation

We have an integral of a function that depends on the variable t (with its corresponding differential of t), that goes from a constant “a” to the variable “x”, which we will call F(x):

If we derive all that integral with respect to x, the result is the same function that depended on t in the integral, but now depending on x:

In other words, the derivative of the integral of a function is that function itself. What shows that deriving is the inverse operation to integrate.

The application of this theorem is very simple, since we only have to replace the t with the x in the function we are integrating.

## Solved exercises of the fundamental theorem of the calculation

For example, find the derivative of the next integral using the fundamental theorem:

We call that expression F(x):

We are being asked to calculate the derivative of F(x), i.e. F'(x), which is the derivative of the integral with respect to x:

So let’s solve this derivative here:

Applying the formula of the fundamental calculus theorem:

The function in this case f(t) is:

To find f(x), we only have to replace the x with the t in the function f(t) that is inside the integral. Replacing t with x leaves f(x):

Therefore, the derivative of the integral is:

Which is the same as saying that the derivative of F(x) is equal to f(x):

You must be careful because in the formula of the fundamental theorem, the integral goes from the constant “a” to the variable “x”, therefore, if the integral is written the opposite way, you must turn the integral over and put a minus sign in front of it so that its result does not vary.

Let’s see this with an example: Find the derivative of the next integral using the fundamental theorem:

We are asked to derive with respect to x this integral:

We see that the integral goes from x to -2 and to be able to apply the formula of the fundamental theorem, it should go from -2 to x, therefore, we turn the integral and place a minus sign in front of it so as not to vary its result (according to the properties of the integrals):

Therefore, the derivative of the original integral is replaced by the derivative we have turned around and has a minus sign in front of it:

And that minus sign can be taken out of the integral:

Now we can apply the formula of the fundamental theorem, bearing in mind that we have a minus sign in front of us:

f(t) is:

We replace x with t to get f(x):

So the derivative of the integral, taking into account the minus sign that leads ahead is:

As the derivative of the original integral is equal to the derivative of the integral we have turned around and with the minus sign in front:

The result of the derivative of the original integral is:

## Fundamental theorem formula to derive integrals ranging from a constant to a function

So far we’ve seen how to derive integrals ranging from a constant to the variable x, but how do you apply the fundamental theorem of calculus when you want to derive integrals ranging from a constant to a function?

I’m going to explain how to apply the formula of the fundamental theorem to derive integrals ranging from a constant to a function, instead of going from a constant to the x variable, that is, I’m going to explain how to derive integrals like this one, which we call F(x):

applying the fundamental theorem.

In this case, the derivative of F(x) with respect to x will be equal to the function f(u(x)), by the derivative of u(x). The result is the same function that depended on t, but now depending on u(x) and multiplied by the derivative of that function u(x):

Let’s see how to apply this formula with an example: Find the derivative of the following integral:

We are going to calculate the derivative of this integral with respect to x using the fundamental theorem:

We identify u(x) that it is:

We get f(u(x)), replacing t with u(x):

And we calculate u'(x) by deriving u(x):

Therefore, the derivative of the original integral, applying the formula of the fundamental theorem is equal to:

## Fundamental theorem formula to derive integrals ranging from one function to another function

Finally I’m going to explain how to apply the formula of the fundamental theorem to derive integrals that go from one function to another function, like this one:

To derive integrals ranging from a function to a function the formula of the fundamental theorem is as follows:

And to understand it better we are going to apply it deriving the following integral:

We want to derive that integral with respect to x:

Let’s define each of the components that appear in the formula. v(x) and u(x) are:

To obtain f(u(x)) we substitute the t in f(t) with u(x):

To obtain f(v(x)) we replace the t in f(t) with v(x):

The derivatives of v(x) and u(x) are:

We substituted everything in the formula:

And if we continue operating, the result is: