Integration Using Partial Fractions
( i) If f( x) and g( x) are two polynomials, then f( x) / g( x) defines a rational algebraic function of x. If degree of f( x) < degree of g( x), then f( x) / g( x) is called a proper rational function.
( ii) If degree of f( x) ≥ degree of g( x), then f( x) / g( x) is called an improper g( x) rational function.
( iii) If f( x) / g( x) isan improper rational function, then we divide f( x) by g( x) g( x) and convert it into a proper rational function as f( x) / g( x) = φ( x) + h( x) / g( x).
( iv) Any proper rational function f( x) / g( x) can be expressed as the sum of rational functions each having a simple factor of g( x). Each such fraction is called a partial fraction and the process of obtaining them, is called the resolution or decomposition of f( x) / g( x) partial fraction.