A rational function f (x) can be represented by the ratio of two polynomials,

.

A rational function is said to be a proper fraction, if the degree of the polynomial P (x) is less than the degree of the polynomial Q (x).

For example, the expressions

.

are the rational functions, the last two of which are the proper fractions, while the expression

is not a rational function.

Fractions of the form

   and  

(with integer ) are called the partial fractions, where the quadratic polynomial  x2 + px + q  is assumed to be irreducible.

A proper fraction can be decomposed into a sum of partial fractions. The corresponding method is called “Decomposition of a Rational Function into the Sum of Partial Fractions” or in short, “Partial Fraction Decomposition”.

If  f (x)  is not a proper fraction then it is necessary to perform the polynomial long division in order to represent   f (x)  by the sum of a polynomial and a proper fraction.

The integration of a polynomial is a trivial problem. Therefore, the integration of a rational function is reduced to the integration of proper fractions.

Indefinite Integrals Definite Integrals Differential Equations