Rational integrals with different real roots. Exercises resolved.

Do you know how to integrate rational functions? Do you know which integration method to use?

Not only is it important to know how to integrate rational functions by performing the integration method correctly, but it is even more important to know which integration method to apply.

One of your problems may be that when you see an integral, you don’t know which integration method to use, even if you know how to apply the method.

Next I am going to explain not only how to integrate rational functions, but also how to know when to apply these methods, or rather, when not to apply them. But let’s start at the beginning:

What is a rational function?

A rational function is one that has the form of a fraction, whose numerator and denominator are polynomials. They have this form:

How to integrate rational functions

These are some examples of rational functions:

integrals of rational functions

The degree of polynomial of the numerator P(x), can be zero, as for example:

integration of rational functions

However, the minimum degree of the polynomial of the denominator Q(x) must be greater than or equal to 1, since if it were zero, we would not have any fraction, but a polynomial:

rational integrals with complex roots

Having said that, I’m going to explain how to integrate rational functions in the next section.

How to integrate rational functions step by step

To know how to integrate rational functions we must pay attention to the numerator and denominator degrees. If we have this integral:

integration of rational fractions

We can distinguish two groups:

1 – That the polynomial degree of the numerator is less than the polynomial degree of the denominator

function integration

And within this group we have three cases:

1.1 – Q(x) has distinct real roots

1.2 – Q(x) has multiple real roots

1.3 – Q(x) has complex roots

The other group that we can distinguish between integrals of rational functions is:

2 – That the degree of the polynomial of the numerator is greater than or equal to the degree of the polynomial of the denominator

How to make integrals step by step

Depending on what type it is, one method or another will be applied.

Now I’m going to focus on explaining how to integrate rational functions that are in the first group (grade P(x) < grade Q(x)) and within them those that have different real roots.

Let’s explain it with an example, step by step.

How to solve rational integrals with different real roots in the denominator

As I mentioned before, I am going to make an example when the polynomial of the denominator has different real roots.

For example, we have this integral:

rational integrals

How do I know which integration method to use? The first step is to identify what kind of integral it is.

We have to discard integration methods. It cannot be solved with any immediate integral of a simple function, since in the denominator we have a function with more than one term.

The immediate integral of a compound function that can be most similar is the logarithmic type:

integral of rational polynomials

But if we try to solve it with this immediate integral, we must make the following variable change:

integrals like knowing which method to use

And we have no terms to replace dt, nor can we add them.

It would have been possible to apply this immediate integral if we had to integrate this function:

integration rational functions

That if you realize, it is also the integral of a rational function, but it is resolved with the method of the immediate integral of a compound function.

Therefore, once the other integration methods have been discarded, it is when we apply the method I am going to teach you. We therefore have the integral of a rational function:

how to solve rational integrals step by step

Of the form:

integrate fractions

Dwhere the grade of the denominator polynomial is greater than the grade of the numerator polynomial:

integration of rational functions

In order to integrate this fraction, we have to decompose it in a sum of simple fractions, in which the numerator degree will be 0, that is, we will have a number and the denominator degree will be 1, so that each one of these simple fractions can be integrated later with immediate integrals of logarithmic type:

rational integrals

We are going to decompose the fraction into a sum of simple fractions:

First, we have to factor the polynomial of the denominator.

In this case it is a polynomial of degree 2. To decompose it we find its roots (solutions of the second degree equation) and for this we equal it to 0 and solve the equation:

how to integrate a rational function step by step

Whose solutions are:

How to solve rational function integrals

At this point is where we know that the denominator has different real roots.

Each root of the polynomial can be written in the form of a binomial (x-a), where a is the value of each root and in addition, any one polynomial can be written as the product of all its binomials (x-a). Therefore:

calculate integrals of rational functions

If the denominator polynomial was grade 3 or higher, other methods would have to be used to decompose it.

Now, the integral fraction can be written as a sum of fractions whose denominators are each of the denominator factors:

rational integrals step by step

And the numerators of each fraction are two constants that we do not know:

integral rational

Now let’s calculate the value of A and B.

We have a sum of two fractions with different denominator. In order to add them we have to find a common denominator, which will be the multiplication of the denominators:

integration method by simple fractions

And to find each new numerator, the common denominator is divided by the denominator of each initial fraction and the result is multiplied by the numerator of the initial fraction:

How to integrate a division of polynomials

And we can write this as a single fraction:

integration by rational fractions

And here we have come to the following conclusion: The original fraction is equal to the fraction we have just calculated:

fraction integrals

Therefore, as we know the denominators are the same:

integrate a fraction

The numerators will also be the same:

integration rational functions

And we keep this last expression. We are going to give values to x to calculate A and B.

The values that must be given are the roots of the polynomial, since it will cause A or B to be annulled and we know its result. Then:

integral of a rational function

Now that we know the values of A and B, we substitute them in this expression we had at the beginning:

rational integrals exercises

integration of rational functions exercises solved

And we already have the fraction broken down into a sum of simple fractions. To understand what we have done so far, you need to master how to operate with polynomials and algebraic fractions.

We continue with the integral. Now, we write the integral with our new simple fractions:

integration of rational functions by partial fractions

And each of these integrals are immediate integrals of logarithmic type:

integration of rational functions with real roots in the denominatorWe solve them and add the constant:

integration of rational functions by partial fractions resolved exercises

Do you master the immediate integrals? This is another of the things you need to master and which I can also explain to you step by step.

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