Integration of Rational Functions Using Partial Fractions

Rational functions are defined as the ratio of one polynomial to another. They are generally classified into two types—proper and improper.

For a proper rational function, the degree of the numerator is less than that of the denominator. Here, the partial fraction decomposition simplifies the function.

Next, multiply both sides by the denominator to remove fractions. Then, putting an appropriate value for x gives values of the unknowns.

Integrating both sides gives the solution in terms of logarithmic terms and a constant of integration.

For improper rational functions, where the numerator’s degree is greater than or equal to the denominator’s, polynomial long division is applied first.

This process separates the function into two parts: a polynomial term and a proper fraction.

Each part is integrated separately: the polynomials are solved with power rules, while the proper fraction is simplified directly.

The polynomial integrals give algebraic expressions, while the proper fraction produces a logarithmic term plus a constant.

The integration of rational functions, either using partial fractions or divisions, reduces complex functions into sums of simpler integrals.