Integration of Rational Functions Using Partial Fractions

Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.

A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The process involves expressing the integrand with unknown constants, multiplying through by the denominator to eliminate fractions, and solving for the constants by substitution or by matching coefficients. Once decomposed, each term can be integrated directly, typically producing logarithmic expressions along with a constant of integration.

An improper rational function occurs when the degree of the numerator is greater than or equal to that of the denominator. Before integration, polynomial long division is applied to separate the function into a polynomial part and a proper rational fraction. This step reduces the complexity of the integrand and prepares it for further analysis.

The polynomial term obtained from the division is integrated using standard power rules, resulting in algebraic expressions. The remaining proper fraction is then integrated using partial fraction decomposition, yielding logarithmic terms. By applying these techniques in sequence, the integration of rational functions is reduced to a sum of simpler integrals, making the overall process systematic and efficient.