First fundamental theorem of integral calculus. Exercises solved step by step

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.

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):

fundamental theorem of the calculation resolved exercises

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:

primer teorema fundamental del calculo ejercicios resueltos

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.

solved exercises of the fundamental theorem of calculation

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:

ejercicios teorema fundamental del cálculo

We call that expression F(x):

ejercicios de teorema fundamental del cálculo

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:

fundamental theorem of the calculation exercises

So let’s solve this derivative here:

exercises of the fundamental theorem of calculation

Applying the formula of the fundamental calculus theorem:

fundamental calculus theorem

The function in this case f(t) is:

fundamental theorem of integral calculus exercises

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):

primera teorema fundamental del cálculo

Therefore, the derivative of the integral is:

fundamental theorem of integral calculus

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

exercises resolved fundamental theorem of calculation

ejercicios resueltos de teorema fundamental del calculo

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:

fundamental theorem

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

What is the fundamental theorem of calculation used for

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):

fundamental theorem of the calculation definition

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:

examples of the fundamental theorem of the calculation

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

fundamental theorem of calculation formula

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

First fundamental theorem of the calculation resolved exercises

f(t) is:

theorem of integral calculus

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

teorema fundamental de calculo

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

the first fundamental theorem of the calculation

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:

fundamental theorem of integral calculus

The result of the derivative of the original integral is:

fundamental calculus theorem

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):

ejercicios teorema fundamental del cálculo resueltos

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):

fundamental theorem of integral calculation resolved exercises

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

fundamental theorem of calculation examples

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

fundamental theorem of the calculation of solved examples

We identify u(x) that it is:

ejercicios teorema fundamental del cálculo

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

fundamental theorem of integral calculation

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

fundamental theorem of the calculation exercises

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

application of the fundamental theorem of calculation

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:

fundamental theorem of the defined integral calculation

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

which is the fundamental theorem of integral calculation

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

which is the fundamental theorem of the calculation

We want to derive that integral with respect to x:

What is the fundamental theorem of the calculation

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

fundamental theorem of the calculation formula

to serve the fundamental theorem of calculation

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

theorems of the defined integral

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

fundamental theorems of integral calculation

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

fundamental theorem of integral calculation exercises resolved

examples of the fundamental theorem of calculation

We substituted everything in the formula:

fundamental theorem of calculation solved examples

And if we continue operating, the result is:

solved examples of the fundamental theorem of integral calculation

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