Integration of Rational Fractions | Techniques of Integration

Partial Fraction

Functions of x that can be expressed in the form P(x)/Q(x), where both P(x) and Q(x) are polynomials of x, is known as rational fraction. A rational fraction is known to be a proper fraction if the degree of P(x) is less than the degree of Q(x). Example of proper fraction is...
 

$\dfrac{2x^2 + 4x - 5}{5x^3 + 6x^2 -2x -1}$

 
A rational fraction is said to be an improper fraction if the degree of P(x) is greater than or equal to the degree of Q(x). Examples are...
 

$\dfrac{3x^2 - 2x + 1}{2x^2 + 6}\,\,$ and $\,\,\dfrac{4x^2 - 2x + 3}{3x + 2}$

 
Improper fraction may be expressed as the sum of a polynomial and a proper fraction. For example:
 

$\dfrac{12x^2 - 13x - 9}{4x - 7} = 3x + 2 + \dfrac{5}{4x - 7}$

 
Proper fraction such as $\,\, \dfrac{x - 4}{2x^2 - 4x} \,\,$ can be expressed as the sum of partial fraction, provided that the denominator will factorized.
 
Integration of any rational fraction depends essentially on the integration of a proper fraction by expressing it into a sum of partial fractions. There are four cases that may arise in dealing with integrand involving proper fraction.
 

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