How to solve fractional integrals

How to solve fractional integrals? Here are the methods explained, step by step, and the examples to solve these types of integrals in a systematic and efficient way.

Introduction to Fractional Integrals

Fractional integrals are a class of integrals involving algebraic fractions, that is, functions of the type P(X)/Q(x) , where P(x) and Q(x) are polynomials. These integrals can be solved through a specific procedure that includes the decomposition of the denominator into first-degree polynomials .

By following the steps in this guide, you will be able to learn how to solve fractional integrals.

Steps to follow to solve fractional integrals

Before we get into practical examples, it is important to understand the four basic steps needed to solve a fractional integral. Here is what you need to do:

1. Algebraic division : if the degree of the numerator is greater than or equal to the degree of the denominator, we start with the polynomial division between P(x)  and Q(x) ;
2. Factorization of the denominator : the denominator should be factored as a product of first degree polynomials, if possible;
3. Decomposition into simple fractions : the resulting fraction, if necessary, must be written as the sum of simpler fractions;
4. Separate integration : finally, the fractions obtained are integrated separately thanks to the linearity property of the integral.

Ways to solve fractional integrals

When you have a ratio of two polynomials  f(x) = P(x)/Q(x) , there are four main approaches to calculating the integral, depending on the form of the numerator and denominator.

1. Case 1. If the numerator is the derivative of the denominator, the integral is calculated using the natural logarithm, that is f P'(x)/P(x)dx = ln |P(x)| + C .
2. Case 2. If the numerator is a constant and the denominator is a first degree polynomial, we apply the formula for the integral of a rational function of the type f 1/(ax + b)dx = 1/a $\ln |ax+b|+C$ .
3. Case 3. If the numerator is a constant and the denominator is a square of the type (x² + a²) , with the power n = -2 , the formula for the arctangent integral is used, that is, f (1/x²+a²)dx = 1/a arctan (x/a) + C.
4. Case 4. If the numerator is of the first degree and the denominator is a second degree trinomial, techniques for completing the square and/or partial fractions are applied.