Primitive of a function and properties of indefinite integrals

You know what the primitive part of a function is?

Next, I will explain to you what is the primitive of a function, to better understand what is an indefinite integral and we will study the properties of the indefinite integrals, which are fundamental to learn how to solve integrals.

Let’s go over there!

Primitive of a function

What is the primitive of a function?

The primitive of a function is the function from which a derived function proceeds.

A function f'(x) is obtained by deriving a function F(x), for example:

primitiva de una funcion

Then, we will say that F(x) is primitive from f'(x) and to pass from F(x) to f'(x), we must derive f'(x).

What if we have directly f'(x) and we want to find its primitive F(x)?

That is, we want to find the function from which that derived function comes.

An example could be this, in which we start from the derived function and arrive at its primitive one:

properties of integrals

We have found a function (x³) that if we derive it, we return to the function f'(x). She’s the primitive of the show.

However, the following functions are also primitive to the previous function f'(x), since if we derive them, we get exactly the same function f'(x):

properties of undefined integrals

properties of the indefinite integral

The only thing that differentiates these primitives is the number that is added at the end. We will call this number constant and this constant can take any value, so a function has infinite primitives.

The set of all primitives is the indefinite integral and is written as follows:

properties of undefined integrals

In the case of the above example it would be:

which is the primitive of a function

By adding the constant to the primitive, we are encompassing all the possible primitives of the function.

This process opposed to derivation is known as integration.

Properties of the indefinite integrals

We will now see what properties indefinite integrals have, which we will use to simplify the calculations when solving any type of integral.

Property 1

If we have a constant that is multiplying to a function, we can take the constant out of the integral:

funcion primitiva

Property 2

The integral of the addition or subtraction of 2 or more functions is equal to the addition or subtraction of their integrals:

primitive functions

Be very careful with this property because it is not extensible for integrals that have multiplication or division of functions.

The integral of the multiplication of two functions is not equal to the multiplication of their integrals:

que es una primitiva de una funcion

In the same way, the integral of the division of two functions is not the same as the division of the integrals:

the primitive of a function